statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
continuous_of_real_cpow_const {y : ℂ} (hs : 0 < y.re) : continuous (λ x, x ^ y : ℝ → ℂ) | continuous_iff_continuous_at.mpr (λ x, continuous_at_of_real_cpow_const x y (or.inl hs)) | lemma | complex.continuous_of_real_cpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_rpow {x : ℝ≥0} {y : ℝ} (h : x ≠ 0 ∨ 0 < y) :
continuous_at (λp:ℝ≥0×ℝ, p.1^p.2) (x, y) | begin
have : (λp:ℝ≥0×ℝ, p.1^p.2) = real.to_nnreal ∘ (λp:ℝ×ℝ, p.1^p.2) ∘ (λp:ℝ≥0 × ℝ, (p.1.1, p.2)),
{ ext p,
rw [coe_rpow, real.coe_to_nnreal _ (real.rpow_nonneg_of_nonneg p.1.2 _)],
refl },
rw this,
refine continuous_real_to_nnreal.continuous_at.comp (continuous_at.comp _ _),
{ apply real.continuous_... | lemma | nnreal.continuous_at_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at",
"continuous_at.comp",
"continuous_fst",
"continuous_snd",
"nnreal.coe_eq_zero",
"real.coe_to_nnreal",
"real.continuous_at_rpow",
"real.rpow_nonneg_of_nonneg",
"real.to_nnreal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_pow_one_div_le (x : ℝ≥0) {y : ℝ≥0} (hy : 1 < y) :
∀ᶠ (n : ℕ) in at_top, x ^ (1 / n : ℝ) ≤ y | begin
obtain ⟨m, hm⟩ := add_one_pow_unbounded_of_pos x (tsub_pos_of_lt hy),
rw [tsub_add_cancel_of_le hy.le] at hm,
refine eventually_at_top.2 ⟨m + 1, λ n hn, _⟩,
simpa only [nnreal.rpow_one_div_le_iff (nat.cast_pos.2 $ m.succ_pos.trans_le hn),
nnreal.rpow_nat_cast] using hm.le.trans (pow_le_pow hy.le (m.le... | lemma | nnreal.eventually_pow_one_div_le | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"add_one_pow_unbounded_of_pos",
"nnreal.rpow_nat_cast",
"nnreal.rpow_one_div_le_iff",
"pow_le_pow",
"tsub_add_cancel_of_le",
"tsub_pos_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.nnrpow {α : Type*} {f : filter α} {u : α → ℝ≥0} {v : α → ℝ} {x : ℝ≥0} {y : ℝ}
(hx : tendsto u f (𝓝 x)) (hy : tendsto v f (𝓝 y)) (h : x ≠ 0 ∨ 0 < y) :
tendsto (λ a, (u a) ^ (v a)) f (𝓝 (x ^ y)) | tendsto.comp (nnreal.continuous_at_rpow h) (hx.prod_mk_nhds hy) | lemma | filter.tendsto.nnrpow | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"filter",
"nnreal.continuous_at_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_rpow_const {x : ℝ≥0} {y : ℝ} (h : x ≠ 0 ∨ 0 ≤ y) :
continuous_at (λ z, z^y) x | h.elim (λ h, tendsto_id.nnrpow tendsto_const_nhds (or.inl h)) $
λ h, h.eq_or_lt.elim
(λ h, h ▸ by simp only [rpow_zero, continuous_at_const])
(λ h, tendsto_id.nnrpow tendsto_const_nhds (or.inr h)) | lemma | nnreal.continuous_at_rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at",
"continuous_at_const",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_rpow_const {y : ℝ} (h : 0 ≤ y) :
continuous (λ x : ℝ≥0, x^y) | continuous_iff_continuous_at.2 $ λ x, continuous_at_rpow_const (or.inr h) | lemma | nnreal.continuous_rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_pow_one_div_le {x : ℝ≥0∞} (hx : x ≠ ∞) {y : ℝ≥0∞} (hy : 1 < y) :
∀ᶠ (n : ℕ) in at_top, x ^ (1 / n : ℝ) ≤ y | begin
lift x to ℝ≥0 using hx,
by_cases y = ∞,
{ exact eventually_of_forall (λ n, h.symm ▸ le_top) },
{ lift y to ℝ≥0 using h,
have := nnreal.eventually_pow_one_div_le x (by exact_mod_cast hy : 1 < y),
refine this.congr (eventually_of_forall $ λ n, _),
rw [coe_rpow_of_nonneg x (by positivity : 0 ≤ (1... | lemma | ennreal.eventually_pow_one_div_le | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"le_top",
"lift",
"nnreal.eventually_pow_one_div_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_rpow_const_of_pos {x : ℝ≥0∞} {y : ℝ} (h : 0 < y) :
continuous_at (λ a : ℝ≥0∞, a ^ y) x | begin
by_cases hx : x = ⊤,
{ rw [hx, continuous_at],
convert tendsto_rpow_at_top h,
simp [h] },
lift x to ℝ≥0 using hx,
rw continuous_at_coe_iff,
convert continuous_coe.continuous_at.comp
(nnreal.continuous_at_rpow_const (or.inr h.le)) using 1,
ext1 x,
simp [coe_rpow_of_nonneg _ h.le]
end | lemma | ennreal.continuous_at_rpow_const_of_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous_at",
"lift",
"nnreal.continuous_at_rpow_const",
"tendsto_rpow_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_rpow_const {y : ℝ} : continuous (λ a : ℝ≥0∞, a ^ y) | begin
apply continuous_iff_continuous_at.2 (λ x, _),
rcases lt_trichotomy 0 y with hy|rfl|hy,
{ exact continuous_at_rpow_const_of_pos hy },
{ simp only [rpow_zero], exact continuous_at_const },
{ obtain ⟨z, hz⟩ : ∃ z, y = -z := ⟨-y, (neg_neg _).symm⟩,
have z_pos : 0 < z, by simpa [hz] using hy,
simp_r... | lemma | ennreal.continuous_rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"continuous",
"continuous_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_const_mul_rpow_nhds_zero_of_pos {c : ℝ≥0∞} (hc : c ≠ ∞) {y : ℝ} (hy : 0 < y) :
tendsto (λ x : ℝ≥0∞, c * x ^ y) (𝓝 0) (𝓝 0) | begin
convert ennreal.tendsto.const_mul (ennreal.continuous_rpow_const.tendsto 0) _,
{ simp [hy] },
{ exact or.inr hc }
end | lemma | ennreal.tendsto_const_mul_rpow_nhds_zero_of_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"ennreal.tendsto.const_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.ennrpow_const {α : Type*} {f : filter α} {m : α → ℝ≥0∞} {a : ℝ≥0∞} (r : ℝ)
(hm : tendsto m f (𝓝 a)) :
tendsto (λ x, (m x) ^ r) f (𝓝 (a ^ r)) | (ennreal.continuous_rpow_const.tendsto a).comp hm | lemma | filter.tendsto.ennrpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/continuity.lean | [
"analysis.special_functions.pow.asymptotics"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at_cpow {p : ℂ × ℂ} (hp : 0 < p.1.re ∨ p.1.im ≠ 0) :
has_strict_fderiv_at (λ x : ℂ × ℂ, x.1 ^ x.2)
((p.2 * p.1 ^ (p.2 - 1)) • continuous_linear_map.fst ℂ ℂ ℂ +
(p.1 ^ p.2 * log p.1) • continuous_linear_map.snd ℂ ℂ ℂ) p | begin
have A : p.1 ≠ 0, by { intro h, simpa [h, lt_irrefl] using hp },
have : (λ x : ℂ × ℂ, x.1 ^ x.2) =ᶠ[𝓝 p] (λ x, exp (log x.1 * x.2)),
from ((is_open_ne.preimage continuous_fst).eventually_mem A).mono
(λ p hp, cpow_def_of_ne_zero hp _),
rw [cpow_sub _ _ A, cpow_one, mul_div_left_comm, mul_smul, mul... | lemma | complex.has_strict_fderiv_at_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"continuous_fst",
"continuous_linear_map.fst",
"continuous_linear_map.snd",
"div_eq_mul_inv",
"exp",
"has_strict_fderiv_at",
"has_strict_fderiv_at.congr_of_eventually_eq",
"has_strict_fderiv_at_snd",
"mul_div_left_comm",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at_cpow' {x y : ℂ} (hp : 0 < x.re ∨ x.im ≠ 0) :
has_strict_fderiv_at (λ x : ℂ × ℂ, x.1 ^ x.2)
((y * x ^ (y - 1)) • continuous_linear_map.fst ℂ ℂ ℂ +
(x ^ y * log x) • continuous_linear_map.snd ℂ ℂ ℂ) (x, y) | @has_strict_fderiv_at_cpow (x, y) hp | lemma | complex.has_strict_fderiv_at_cpow' | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"continuous_linear_map.fst",
"continuous_linear_map.snd",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_const_cpow {x y : ℂ} (h : x ≠ 0 ∨ y ≠ 0) :
has_strict_deriv_at (λ y, x ^ y) (x ^ y * log x) y | begin
rcases em (x = 0) with rfl|hx,
{ replace h := h.neg_resolve_left rfl,
rw [log_zero, mul_zero],
refine (has_strict_deriv_at_const _ 0).congr_of_eventually_eq _,
exact (is_open_ne.eventually_mem h).mono (λ y hy, (zero_cpow hy).symm) },
{ simpa only [cpow_def_of_ne_zero hx, mul_one]
using ((h... | lemma | complex.has_strict_deriv_at_const_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"em",
"has_strict_deriv_at",
"has_strict_deriv_at_const",
"has_strict_deriv_at_id",
"mul_one",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_cpow {p : ℂ × ℂ} (hp : 0 < p.1.re ∨ p.1.im ≠ 0) :
has_fderiv_at (λ x : ℂ × ℂ, x.1 ^ x.2)
((p.2 * p.1 ^ (p.2 - 1)) • continuous_linear_map.fst ℂ ℂ ℂ +
(p.1 ^ p.2 * log p.1) • continuous_linear_map.snd ℂ ℂ ℂ) p | (has_strict_fderiv_at_cpow hp).has_fderiv_at | lemma | complex.has_fderiv_at_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"continuous_linear_map.fst",
"continuous_linear_map.snd",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.cpow (hf : has_strict_fderiv_at f f' x)
(hg : has_strict_fderiv_at g g' x) (h0 : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_strict_fderiv_at (λ x, f x ^ g x)
((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * log (f x)) • g') x | by convert (@has_strict_fderiv_at_cpow ((λ x, (f x, g x)) x) h0).comp x (hf.prod hg) | lemma | has_strict_fderiv_at.cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.const_cpow (hf : has_strict_fderiv_at f f' x) (h0 : c ≠ 0 ∨ f x ≠ 0) :
has_strict_fderiv_at (λ x, c ^ f x) ((c ^ f x * log c) • f') x | (has_strict_deriv_at_const_cpow h0).comp_has_strict_fderiv_at x hf | lemma | has_strict_fderiv_at.const_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.cpow (hf : has_fderiv_at f f' x) (hg : has_fderiv_at g g' x)
(h0 : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_fderiv_at (λ x, f x ^ g x)
((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * log (f x)) • g') x | by convert (@complex.has_fderiv_at_cpow ((λ x, (f x, g x)) x) h0).comp x (hf.prod hg) | lemma | has_fderiv_at.cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"complex.has_fderiv_at_cpow",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.const_cpow (hf : has_fderiv_at f f' x) (h0 : c ≠ 0 ∨ f x ≠ 0) :
has_fderiv_at (λ x, c ^ f x) ((c ^ f x * log c) • f') x | (has_strict_deriv_at_const_cpow h0).has_deriv_at.comp_has_fderiv_at x hf | lemma | has_fderiv_at.const_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_deriv_at.comp_has_fderiv_at",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.cpow (hf : has_fderiv_within_at f f' s x)
(hg : has_fderiv_within_at g g' s x) (h0 : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_fderiv_within_at (λ x, f x ^ g x)
((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * log (f x)) • g') s x | by convert (@complex.has_fderiv_at_cpow ((λ x, (f x, g x)) x) h0).comp_has_fderiv_within_at x
(hf.prod hg) | lemma | has_fderiv_within_at.cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"complex.has_fderiv_at_cpow",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.const_cpow (hf : has_fderiv_within_at f f' s x) (h0 : c ≠ 0 ∨ f x ≠ 0) :
has_fderiv_within_at (λ x, c ^ f x) ((c ^ f x * log c) • f') s x | (has_strict_deriv_at_const_cpow h0).has_deriv_at.comp_has_fderiv_within_at x hf | lemma | has_fderiv_within_at.const_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_deriv_at.comp_has_fderiv_within_at",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.cpow (hf : differentiable_at ℂ f x) (hg : differentiable_at ℂ g x)
(h0 : 0 < (f x).re ∨ (f x).im ≠ 0) :
differentiable_at ℂ (λ x, f x ^ g x) x | (hf.has_fderiv_at.cpow hg.has_fderiv_at h0).differentiable_at | lemma | differentiable_at.cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.const_cpow (hf : differentiable_at ℂ f x) (h0 : c ≠ 0 ∨ f x ≠ 0) :
differentiable_at ℂ (λ x, c ^ f x) x | (hf.has_fderiv_at.const_cpow h0).differentiable_at | lemma | differentiable_at.const_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.cpow (hf : differentiable_within_at ℂ f s x)
(hg : differentiable_within_at ℂ g s x) (h0 : 0 < (f x).re ∨ (f x).im ≠ 0) :
differentiable_within_at ℂ (λ x, f x ^ g x) s x | (hf.has_fderiv_within_at.cpow hg.has_fderiv_within_at h0).differentiable_within_at | lemma | differentiable_within_at.cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.const_cpow (hf : differentiable_within_at ℂ f s x)
(h0 : c ≠ 0 ∨ f x ≠ 0) :
differentiable_within_at ℂ (λ x, c ^ f x) s x | (hf.has_fderiv_within_at.const_cpow h0).differentiable_within_at | lemma | differentiable_within_at.const_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aux :
((g x * f x ^ (g x - 1)) • (1 : ℂ →L[ℂ] ℂ).smul_right f' +
(f x ^ g x * log (f x)) • (1 : ℂ →L[ℂ] ℂ).smul_right g') 1 =
g x * f x ^ (g x - 1) * f' + f x ^ g x * log (f x) * g' | by simp only [algebra.id.smul_eq_mul, one_mul, continuous_linear_map.one_apply,
continuous_linear_map.smul_right_apply, continuous_linear_map.add_apply, pi.smul_apply,
continuous_linear_map.coe_smul'] | lemma | aux | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"algebra.id.smul_eq_mul",
"continuous_linear_map.add_apply",
"continuous_linear_map.coe_smul'",
"continuous_linear_map.one_apply",
"continuous_linear_map.smul_right_apply",
"one_mul",
"pi.smul_apply"
] | A private lemma that rewrites the output of lemmas like `has_fderiv_at.cpow` to the form
expected by lemmas like `has_deriv_at.cpow`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_deriv_at.cpow (hf : has_strict_deriv_at f f' x) (hg : has_strict_deriv_at g g' x)
(h0 : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_strict_deriv_at (λ x, f x ^ g x)
(g x * f x ^ (g x - 1) * f' + f x ^ g x * log (f x) * g') x | by simpa only [aux] using (hf.cpow hg h0).has_strict_deriv_at | lemma | has_strict_deriv_at.cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"aux",
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.const_cpow (hf : has_strict_deriv_at f f' x) (h : c ≠ 0 ∨ f x ≠ 0) :
has_strict_deriv_at (λ x, c ^ f x) (c ^ f x * log c * f') x | (has_strict_deriv_at_const_cpow h).comp x hf | lemma | has_strict_deriv_at.const_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complex.has_strict_deriv_at_cpow_const (h : 0 < x.re ∨ x.im ≠ 0) :
has_strict_deriv_at (λ z : ℂ, z ^ c) (c * x ^ (c - 1)) x | by simpa only [mul_zero, add_zero, mul_one]
using (has_strict_deriv_at_id x).cpow (has_strict_deriv_at_const x c) h | lemma | complex.has_strict_deriv_at_cpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_strict_deriv_at",
"has_strict_deriv_at_const",
"has_strict_deriv_at_id",
"mul_one",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.cpow_const (hf : has_strict_deriv_at f f' x)
(h0 : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_strict_deriv_at (λ x, f x ^ c) (c * f x ^ (c - 1) * f') x | (complex.has_strict_deriv_at_cpow_const h0).comp x hf | lemma | has_strict_deriv_at.cpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"complex.has_strict_deriv_at_cpow_const",
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.cpow (hf : has_deriv_at f f' x) (hg : has_deriv_at g g' x)
(h0 : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_deriv_at (λ x, f x ^ g x) (g x * f x ^ (g x - 1) * f' + f x ^ g x * log (f x) * g') x | by simpa only [aux] using (hf.has_fderiv_at.cpow hg h0).has_deriv_at | lemma | has_deriv_at.cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"aux",
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.const_cpow (hf : has_deriv_at f f' x) (h0 : c ≠ 0 ∨ f x ≠ 0) :
has_deriv_at (λ x, c ^ f x) (c ^ f x * log c * f') x | (has_strict_deriv_at_const_cpow h0).has_deriv_at.comp x hf | lemma | has_deriv_at.const_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_deriv_at",
"has_deriv_at.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.cpow_const (hf : has_deriv_at f f' x) (h0 : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_deriv_at (λ x, f x ^ c) (c * f x ^ (c - 1) * f') x | (complex.has_strict_deriv_at_cpow_const h0).has_deriv_at.comp x hf | lemma | has_deriv_at.cpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"complex.has_strict_deriv_at_cpow_const",
"has_deriv_at",
"has_deriv_at.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.cpow (hf : has_deriv_within_at f f' s x)
(hg : has_deriv_within_at g g' s x) (h0 : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_deriv_within_at (λ x, f x ^ g x)
(g x * f x ^ (g x - 1) * f' + f x ^ g x * log (f x) * g') s x | by simpa only [aux] using (hf.has_fderiv_within_at.cpow hg h0).has_deriv_within_at | lemma | has_deriv_within_at.cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"aux",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.const_cpow (hf : has_deriv_within_at f f' s x) (h0 : c ≠ 0 ∨ f x ≠ 0) :
has_deriv_within_at (λ x, c ^ f x) (c ^ f x * log c * f') s x | (has_strict_deriv_at_const_cpow h0).has_deriv_at.comp_has_deriv_within_at x hf | lemma | has_deriv_within_at.const_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_deriv_at.comp_has_deriv_within_at",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.cpow_const (hf : has_deriv_within_at f f' s x)
(h0 : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_deriv_within_at (λ x, f x ^ c) (c * f x ^ (c - 1) * f') s x | (complex.has_strict_deriv_at_cpow_const h0).has_deriv_at.comp_has_deriv_within_at x hf | lemma | has_deriv_within_at.cpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"complex.has_strict_deriv_at_cpow_const",
"has_deriv_at.comp_has_deriv_within_at",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_of_real_cpow {x : ℝ} (hx : x ≠ 0) {r : ℂ} (hr : r ≠ -1) :
has_deriv_at (λ y:ℝ, (y:ℂ) ^ (r + 1) / (r + 1)) (x ^ r) x | begin
rw [ne.def, ←add_eq_zero_iff_eq_neg, ←ne.def] at hr,
rcases lt_or_gt_of_ne hx.symm with hx | hx,
{ -- easy case : `0 < x`
convert (((has_deriv_at_id (x:ℂ)).cpow_const _).div_const (r + 1)).comp_of_real,
{ rw [add_sub_cancel, id.def, mul_one, mul_comm, mul_div_cancel _ hr] },
{ rw [id.def, of_rea... | lemma | has_deriv_at_of_real_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"Iio_mem_nhds",
"exp",
"exp_add",
"filter.eventually_of_mem",
"has_deriv_at",
"has_deriv_at.congr_of_eventually_eq",
"has_deriv_at.scomp",
"has_deriv_at_id",
"has_deriv_at_neg",
"mul_assoc",
"mul_comm",
"mul_div_cancel",
"mul_neg",
"mul_one",
"neg_one_mul",
"nhds",
"ring"
] | Although `λ x, x ^ r` for fixed `r` is *not* complex-differentiable along the negative real
line, it is still real-differentiable, and the derivative is what one would formally expect. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_fderiv_at_rpow_of_pos (p : ℝ × ℝ) (hp : 0 < p.1) :
has_strict_fderiv_at (λ x : ℝ × ℝ, x.1 ^ x.2)
((p.2 * p.1 ^ (p.2 - 1)) • continuous_linear_map.fst ℝ ℝ ℝ +
(p.1 ^ p.2 * log p.1) • continuous_linear_map.snd ℝ ℝ ℝ) p | begin
have : (λ x : ℝ × ℝ, x.1 ^ x.2) =ᶠ[𝓝 p] (λ x, exp (log x.1 * x.2)),
from (continuous_at_fst.eventually (lt_mem_nhds hp)).mono (λ p hp, rpow_def_of_pos hp _),
refine has_strict_fderiv_at.congr_of_eventually_eq _ this.symm,
convert ((has_strict_fderiv_at_fst.log hp.ne').mul has_strict_fderiv_at_snd).exp,... | lemma | real.has_strict_fderiv_at_rpow_of_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"continuous_linear_map.fst",
"continuous_linear_map.snd",
"div_eq_mul_inv",
"exp",
"has_strict_fderiv_at",
"has_strict_fderiv_at.congr_of_eventually_eq",
"has_strict_fderiv_at_snd",
"lt_mem_nhds",
"mul_assoc",
"mul_div_left_comm",
"smul_add",
"smul_smul"
] | `(x, y) ↦ x ^ y` is strictly differentiable at `p : ℝ × ℝ` such that `0 < p.fst`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_fderiv_at_rpow_of_neg (p : ℝ × ℝ) (hp : p.1 < 0) :
has_strict_fderiv_at (λ x : ℝ × ℝ, x.1 ^ x.2)
((p.2 * p.1 ^ (p.2 - 1)) • continuous_linear_map.fst ℝ ℝ ℝ +
(p.1 ^ p.2 * log p.1 - exp (log p.1 * p.2) * sin (p.2 * π) * π) •
continuous_linear_map.snd ℝ ℝ ℝ) p | begin
have : (λ x : ℝ × ℝ, x.1 ^ x.2) =ᶠ[𝓝 p] (λ x, exp (log x.1 * x.2) * cos (x.2 * π)),
from (continuous_at_fst.eventually (gt_mem_nhds hp)).mono (λ p hp, rpow_def_of_neg hp _),
refine has_strict_fderiv_at.congr_of_eventually_eq _ this.symm,
convert ((has_strict_fderiv_at_fst.log hp.ne).mul has_strict_fder... | lemma | real.has_strict_fderiv_at_rpow_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"add_smul",
"continuous_linear_map.fst",
"continuous_linear_map.snd",
"div_eq_mul_inv",
"exp",
"gt_mem_nhds",
"has_strict_fderiv_at",
"has_strict_fderiv_at.congr_of_eventually_eq",
"has_strict_fderiv_at_snd",
"mul_assoc",
"mul_comm",
"ring",
"smul_add",
"smul_smul"
] | `(x, y) ↦ x ^ y` is strictly differentiable at `p : ℝ × ℝ` such that `p.fst < 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at_rpow_of_ne (p : ℝ × ℝ) (hp : p.1 ≠ 0) {n : ℕ∞} :
cont_diff_at ℝ n (λ p : ℝ × ℝ, p.1 ^ p.2) p | begin
cases hp.lt_or_lt with hneg hpos,
exacts [(((cont_diff_at_fst.log hneg.ne).mul cont_diff_at_snd).exp.mul
(cont_diff_at_snd.mul cont_diff_at_const).cos).congr_of_eventually_eq
((continuous_at_fst.eventually (gt_mem_nhds hneg)).mono (λ p hp, rpow_def_of_neg hp _)),
((cont_diff_at_fst.log hpos.ne')... | lemma | real.cont_diff_at_rpow_of_ne | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_at",
"cont_diff_at_const",
"cont_diff_at_snd",
"gt_mem_nhds",
"lt_mem_nhds"
] | The function `λ (x, y), x ^ y` is infinitely smooth at `(x, y)` unless `x = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_at_rpow_of_ne (p : ℝ × ℝ) (hp : p.1 ≠ 0) :
differentiable_at ℝ (λ p : ℝ × ℝ, p.1 ^ p.2) p | (cont_diff_at_rpow_of_ne p hp).differentiable_at le_rfl | lemma | real.differentiable_at_rpow_of_ne | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_at",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.has_strict_deriv_at.rpow {f g : ℝ → ℝ} {f' g' : ℝ} (hf : has_strict_deriv_at f f' x)
(hg : has_strict_deriv_at g g' x) (h : 0 < f x) :
has_strict_deriv_at (λ x, f x ^ g x)
(f' * g x * (f x) ^ (g x - 1) + g' * f x ^ g x * log (f x)) x | begin
convert (has_strict_fderiv_at_rpow_of_pos ((λ x, (f x, g x)) x) h).comp_has_strict_deriv_at _
(hf.prod hg) using 1,
simp [mul_assoc, mul_comm, mul_left_comm]
end | lemma | has_strict_deriv_at.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_strict_deriv_at",
"mul_assoc",
"mul_comm",
"mul_left_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_rpow_const_of_ne {x : ℝ} (hx : x ≠ 0) (p : ℝ) :
has_strict_deriv_at (λ x, x ^ p) (p * x ^ (p - 1)) x | begin
cases hx.lt_or_lt with hx hx,
{ have := (has_strict_fderiv_at_rpow_of_neg (x, p) hx).comp_has_strict_deriv_at x
((has_strict_deriv_at_id x).prod (has_strict_deriv_at_const _ _)),
convert this, simp },
{ simpa using (has_strict_deriv_at_id x).rpow (has_strict_deriv_at_const x p) hx }
end | lemma | real.has_strict_deriv_at_rpow_const_of_ne | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_strict_deriv_at",
"has_strict_deriv_at_const",
"has_strict_deriv_at_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_const_rpow {a : ℝ} (ha : 0 < a) (x : ℝ) :
has_strict_deriv_at (λ x, a ^ x) (a ^ x * log a) x | by simpa using (has_strict_deriv_at_const _ _).rpow (has_strict_deriv_at_id x) ha | lemma | real.has_strict_deriv_at_const_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_strict_deriv_at",
"has_strict_deriv_at_const",
"has_strict_deriv_at_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_const_rpow_of_neg {a x : ℝ} (ha : a < 0) :
has_strict_deriv_at (λ x, a ^ x) (a ^ x * log a - exp (log a * x) * sin (x * π) * π) x | by simpa using (has_strict_fderiv_at_rpow_of_neg (a, x) ha).comp_has_strict_deriv_at x
((has_strict_deriv_at_const _ _).prod (has_strict_deriv_at_id _)) | lemma | real.has_strict_deriv_at_const_rpow_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"exp",
"has_strict_deriv_at",
"has_strict_deriv_at_const",
"has_strict_deriv_at_id"
] | This lemma says that `λ x, a ^ x` is strictly differentiable for `a < 0`. Note that these
values of `a` are outside of the "official" domain of `a ^ x`, and we may redefine `a ^ x`
for negative `a` if some other definition will be more convenient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_rpow_const {x p : ℝ} (h : x ≠ 0 ∨ 1 ≤ p) :
has_deriv_at (λ x, x ^ p) (p * x ^ (p - 1)) x | begin
rcases ne_or_eq x 0 with hx | rfl,
{ exact (has_strict_deriv_at_rpow_const_of_ne hx _).has_deriv_at },
replace h : 1 ≤ p := h.neg_resolve_left rfl,
apply has_deriv_at_of_has_deriv_at_of_ne
(λ x hx, (has_strict_deriv_at_rpow_const_of_ne hx p).has_deriv_at),
exacts [continuous_at_id.rpow_const (or.inr... | lemma | real.has_deriv_at_rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_deriv_at",
"has_deriv_at_of_has_deriv_at_of_ne",
"ne_or_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_rpow_const {p : ℝ} (hp : 1 ≤ p) :
differentiable ℝ (λ x : ℝ, x ^ p) | λ x, (has_deriv_at_rpow_const (or.inr hp)).differentiable_at | lemma | real.differentiable_rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_rpow_const {x p : ℝ} (h : x ≠ 0 ∨ 1 ≤ p) :
deriv (λ x : ℝ, x ^ p) x = p * x ^ (p - 1) | (has_deriv_at_rpow_const h).deriv | lemma | real.deriv_rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"deriv",
"deriv_rpow_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_rpow_const' {p : ℝ} (h : 1 ≤ p) :
deriv (λ x : ℝ, x ^ p) = λ x, p * x ^ (p - 1) | funext $ λ x, deriv_rpow_const (or.inr h) | lemma | real.deriv_rpow_const' | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"deriv",
"deriv_rpow_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_rpow_const_of_ne {x p : ℝ} {n : ℕ∞} (h : x ≠ 0) :
cont_diff_at ℝ n (λ x, x ^ p) x | (cont_diff_at_rpow_of_ne (x, p) h).comp x
(cont_diff_at_id.prod cont_diff_at_const) | lemma | real.cont_diff_at_rpow_const_of_ne | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_at",
"cont_diff_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_rpow_const_of_le {p : ℝ} {n : ℕ} (h : ↑n ≤ p) :
cont_diff ℝ n (λ x : ℝ, x ^ p) | begin
induction n with n ihn generalizing p,
{ exact cont_diff_zero.2 (continuous_id.rpow_const (λ x, by exact_mod_cast or.inr h)) },
{ have h1 : 1 ≤ p, from le_trans (by simp) h,
rw [nat.cast_succ, ← le_sub_iff_add_le] at h,
rw [cont_diff_succ_iff_deriv, deriv_rpow_const' h1],
refine ⟨differentiable_... | lemma | real.cont_diff_rpow_const_of_le | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff",
"cont_diff_succ_iff_deriv",
"nat.cast_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_rpow_const_of_le {x p : ℝ} {n : ℕ} (h : ↑n ≤ p) :
cont_diff_at ℝ n (λ x : ℝ, x ^ p) x | (cont_diff_rpow_const_of_le h).cont_diff_at | lemma | real.cont_diff_at_rpow_const_of_le | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_rpow_const {x p : ℝ} {n : ℕ} (h : x ≠ 0 ∨ ↑n ≤ p) :
cont_diff_at ℝ n (λ x : ℝ, x ^ p) x | h.elim cont_diff_at_rpow_const_of_ne cont_diff_at_rpow_const_of_le | lemma | real.cont_diff_at_rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_rpow_const {x p : ℝ} (hx : x ≠ 0 ∨ 1 ≤ p) :
has_strict_deriv_at (λ x, x ^ p) (p * x ^ (p - 1)) x | cont_diff_at.has_strict_deriv_at'
(cont_diff_at_rpow_const (by rwa nat.cast_one))
(has_deriv_at_rpow_const hx) le_rfl | lemma | real.has_strict_deriv_at_rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_at.has_strict_deriv_at'",
"has_strict_deriv_at",
"le_rfl",
"nat.cast_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.rpow (hf : has_fderiv_within_at f f' s x)
(hg : has_fderiv_within_at g g' s x) (h : 0 < f x) :
has_fderiv_within_at (λ x, f x ^ g x)
((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * log (f x)) • g') s x | (has_strict_fderiv_at_rpow_of_pos (f x, g x) h).has_fderiv_at.comp_has_fderiv_within_at x
(hf.prod hg) | lemma | has_fderiv_within_at.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_fderiv_at.comp_has_fderiv_within_at",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.rpow (hf : has_fderiv_at f f' x) (hg : has_fderiv_at g g' x) (h : 0 < f x) :
has_fderiv_at (λ x, f x ^ g x) ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * log (f x)) • g') x | (has_strict_fderiv_at_rpow_of_pos (f x, g x) h).has_fderiv_at.comp x (hf.prod hg) | lemma | has_fderiv_at.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_fderiv_at",
"has_fderiv_at.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.rpow (hf : has_strict_fderiv_at f f' x)
(hg : has_strict_fderiv_at g g' x) (h : 0 < f x) :
has_strict_fderiv_at (λ x, f x ^ g x)
((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * log (f x)) • g') x | (has_strict_fderiv_at_rpow_of_pos (f x, g x) h).comp x (hf.prod hg) | lemma | has_strict_fderiv_at.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.rpow (hf : differentiable_within_at ℝ f s x)
(hg : differentiable_within_at ℝ g s x) (h : f x ≠ 0) :
differentiable_within_at ℝ (λ x, f x ^ g x) s x | (differentiable_at_rpow_of_ne (f x, g x) h).comp_differentiable_within_at x (hf.prod hg) | lemma | differentiable_within_at.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.rpow (hf : differentiable_at ℝ f x) (hg : differentiable_at ℝ g x)
(h : f x ≠ 0) :
differentiable_at ℝ (λ x, f x ^ g x) x | (differentiable_at_rpow_of_ne (f x, g x) h).comp x (hf.prod hg) | lemma | differentiable_at.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.rpow (hf : differentiable_on ℝ f s) (hg : differentiable_on ℝ g s)
(h : ∀ x ∈ s, f x ≠ 0) :
differentiable_on ℝ (λ x, f x ^ g x) s | λ x hx, (hf x hx).rpow (hg x hx) (h x hx) | lemma | differentiable_on.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.rpow (hf : differentiable ℝ f) (hg : differentiable ℝ g) (h : ∀ x, f x ≠ 0) :
differentiable ℝ (λ x, f x ^ g x) | λ x, (hf x).rpow (hg x) (h x) | lemma | differentiable.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.rpow_const (hf : has_fderiv_within_at f f' s x) (h : f x ≠ 0 ∨ 1 ≤ p) :
has_fderiv_within_at (λ x, f x ^ p) ((p * f x ^ (p - 1)) • f') s x | (has_deriv_at_rpow_const h).comp_has_fderiv_within_at x hf | lemma | has_fderiv_within_at.rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.rpow_const (hf : has_fderiv_at f f' x) (h : f x ≠ 0 ∨ 1 ≤ p) :
has_fderiv_at (λ x, f x ^ p) ((p * f x ^ (p - 1)) • f') x | (has_deriv_at_rpow_const h).comp_has_fderiv_at x hf | lemma | has_fderiv_at.rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.rpow_const (hf : has_strict_fderiv_at f f' x) (h : f x ≠ 0 ∨ 1 ≤ p) :
has_strict_fderiv_at (λ x, f x ^ p) ((p * f x ^ (p - 1)) • f') x | (has_strict_deriv_at_rpow_const h).comp_has_strict_fderiv_at x hf | lemma | has_strict_fderiv_at.rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.rpow_const (hf : differentiable_within_at ℝ f s x)
(h : f x ≠ 0 ∨ 1 ≤ p) :
differentiable_within_at ℝ (λ x, f x ^ p) s x | (hf.has_fderiv_within_at.rpow_const h).differentiable_within_at | lemma | differentiable_within_at.rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.rpow_const (hf : differentiable_at ℝ f x) (h : f x ≠ 0 ∨ 1 ≤ p) :
differentiable_at ℝ (λ x, f x ^ p) x | (hf.has_fderiv_at.rpow_const h).differentiable_at | lemma | differentiable_at.rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.rpow_const (hf : differentiable_on ℝ f s) (h : ∀ x ∈ s, f x ≠ 0 ∨ 1 ≤ p) :
differentiable_on ℝ (λ x, f x ^ p) s | λ x hx, (hf x hx).rpow_const (h x hx) | lemma | differentiable_on.rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.rpow_const (hf : differentiable ℝ f) (h : ∀ x, f x ≠ 0 ∨ 1 ≤ p) :
differentiable ℝ (λ x, f x ^ p) | λ x, (hf x).rpow_const (h x) | lemma | differentiable.rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.const_rpow (hf : has_fderiv_within_at f f' s x) (hc : 0 < c) :
has_fderiv_within_at (λ x, c ^ f x) ((c ^ f x * log c) • f') s x | (has_strict_deriv_at_const_rpow hc (f x)).has_deriv_at.comp_has_fderiv_within_at x hf | lemma | has_fderiv_within_at.const_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_deriv_at.comp_has_fderiv_within_at",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.const_rpow (hf : has_fderiv_at f f' x) (hc : 0 < c) :
has_fderiv_at (λ x, c ^ f x) ((c ^ f x * log c) • f') x | (has_strict_deriv_at_const_rpow hc (f x)).has_deriv_at.comp_has_fderiv_at x hf | lemma | has_fderiv_at.const_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_deriv_at.comp_has_fderiv_at",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.const_rpow (hf : has_strict_fderiv_at f f' x) (hc : 0 < c) :
has_strict_fderiv_at (λ x, c ^ f x) ((c ^ f x * log c) • f') x | (has_strict_deriv_at_const_rpow hc (f x)).comp_has_strict_fderiv_at x hf | lemma | has_strict_fderiv_at.const_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.rpow (hf : cont_diff_within_at ℝ n f s x)
(hg : cont_diff_within_at ℝ n g s x) (h : f x ≠ 0) :
cont_diff_within_at ℝ n (λ x, f x ^ g x) s x | (cont_diff_at_rpow_of_ne (f x, g x) h).comp_cont_diff_within_at x (hf.prod hg) | lemma | cont_diff_within_at.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.rpow (hf : cont_diff_at ℝ n f x) (hg : cont_diff_at ℝ n g x)
(h : f x ≠ 0) :
cont_diff_at ℝ n (λ x, f x ^ g x) x | (cont_diff_at_rpow_of_ne (f x, g x) h).comp x (hf.prod hg) | lemma | cont_diff_at.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.rpow (hf : cont_diff_on ℝ n f s) (hg : cont_diff_on ℝ n g s)
(h : ∀ x ∈ s, f x ≠ 0) :
cont_diff_on ℝ n (λ x, f x ^ g x) s | λ x hx, (hf x hx).rpow (hg x hx) (h x hx) | lemma | cont_diff_on.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.rpow (hf : cont_diff ℝ n f) (hg : cont_diff ℝ n g)
(h : ∀ x, f x ≠ 0) :
cont_diff ℝ n (λ x, f x ^ g x) | cont_diff_iff_cont_diff_at.mpr $
λ x, hf.cont_diff_at.rpow hg.cont_diff_at (h x) | lemma | cont_diff.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.rpow_const_of_ne (hf : cont_diff_within_at ℝ n f s x)
(h : f x ≠ 0) :
cont_diff_within_at ℝ n (λ x, f x ^ p) s x | hf.rpow cont_diff_within_at_const h | lemma | cont_diff_within_at.rpow_const_of_ne | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_within_at",
"cont_diff_within_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.rpow_const_of_ne (hf : cont_diff_at ℝ n f x) (h : f x ≠ 0) :
cont_diff_at ℝ n (λ x, f x ^ p) x | hf.rpow cont_diff_at_const h | lemma | cont_diff_at.rpow_const_of_ne | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_at",
"cont_diff_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.rpow_const_of_ne (hf : cont_diff_on ℝ n f s) (h : ∀ x ∈ s, f x ≠ 0) :
cont_diff_on ℝ n (λ x, f x ^ p) s | λ x hx, (hf x hx).rpow_const_of_ne (h x hx) | lemma | cont_diff_on.rpow_const_of_ne | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.rpow_const_of_ne (hf : cont_diff ℝ n f) (h : ∀ x, f x ≠ 0) :
cont_diff ℝ n (λ x, f x ^ p) | hf.rpow cont_diff_const h | lemma | cont_diff.rpow_const_of_ne | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff",
"cont_diff_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.rpow_const_of_le (hf : cont_diff_within_at ℝ m f s x)
(h : ↑m ≤ p) :
cont_diff_within_at ℝ m (λ x, f x ^ p) s x | (cont_diff_at_rpow_const_of_le h).comp_cont_diff_within_at x hf | lemma | cont_diff_within_at.rpow_const_of_le | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.rpow_const_of_le (hf : cont_diff_at ℝ m f x) (h : ↑m ≤ p) :
cont_diff_at ℝ m (λ x, f x ^ p) x | by { rw ← cont_diff_within_at_univ at *, exact hf.rpow_const_of_le h } | lemma | cont_diff_at.rpow_const_of_le | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_at",
"cont_diff_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.rpow_const_of_le (hf : cont_diff_on ℝ m f s) (h : ↑m ≤ p) :
cont_diff_on ℝ m (λ x, f x ^ p) s | λ x hx, (hf x hx).rpow_const_of_le h | lemma | cont_diff_on.rpow_const_of_le | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.rpow_const_of_le (hf : cont_diff ℝ m f) (h : ↑m ≤ p) :
cont_diff ℝ m (λ x, f x ^ p) | cont_diff_iff_cont_diff_at.mpr $ λ x, hf.cont_diff_at.rpow_const_of_le h | lemma | cont_diff.rpow_const_of_le | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.rpow (hf : has_deriv_within_at f f' s x)
(hg : has_deriv_within_at g g' s x) (h : 0 < f x) :
has_deriv_within_at (λ x, f x ^ g x)
(f' * g x * (f x) ^ (g x - 1) + g' * f x ^ g x * log (f x)) s x | begin
convert (hf.has_fderiv_within_at.rpow hg.has_fderiv_within_at h).has_deriv_within_at using 1,
dsimp, ring
end | lemma | has_deriv_within_at.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_deriv_within_at",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.rpow (hf : has_deriv_at f f' x) (hg : has_deriv_at g g' x) (h : 0 < f x) :
has_deriv_at (λ x, f x ^ g x) (f' * g x * (f x) ^ (g x - 1) + g' * f x ^ g x * log (f x)) x | begin
rw ← has_deriv_within_at_univ at *,
exact hf.rpow hg h
end | lemma | has_deriv_at.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_deriv_at",
"has_deriv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.rpow_const (hf : has_deriv_within_at f f' s x) (hx : f x ≠ 0 ∨ 1 ≤ p) :
has_deriv_within_at (λ y, (f y)^p) (f' * p * (f x) ^ (p - 1)) s x | begin
convert (has_deriv_at_rpow_const hx).comp_has_deriv_within_at x hf using 1,
ring
end | lemma | has_deriv_within_at.rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_deriv_within_at",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.rpow_const (hf : has_deriv_at f f' x) (hx : f x ≠ 0 ∨ 1 ≤ p) :
has_deriv_at (λ y, (f y)^p) (f' * p * (f x)^(p-1)) x | begin
rw ← has_deriv_within_at_univ at *,
exact hf.rpow_const hx
end | lemma | has_deriv_at.rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_deriv_at",
"has_deriv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_rpow_const (hf : differentiable_within_at ℝ f s x) (hx : f x ≠ 0 ∨ 1 ≤ p)
(hxs : unique_diff_within_at ℝ s x) :
deriv_within (λx, (f x) ^ p) s x = (deriv_within f s x) * p * (f x) ^ (p - 1) | (hf.has_deriv_within_at.rpow_const hx).deriv_within hxs | lemma | deriv_within_rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_rpow_const (hf : differentiable_at ℝ f x) (hx : f x ≠ 0 ∨ 1 ≤ p) :
deriv (λx, (f x)^p) x = (deriv f x) * p * (f x)^(p-1) | (hf.has_deriv_at.rpow_const hx).deriv | lemma | deriv_rpow_const | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_one_plus_div_rpow_exp (t : ℝ) :
tendsto (λ (x : ℝ), (1 + t / x) ^ x) at_top (𝓝 (exp t)) | begin
apply ((real.continuous_exp.tendsto _).comp (tendsto_mul_log_one_plus_div_at_top t)).congr' _,
have h₁ : (1:ℝ)/2 < 1 := by linarith,
have h₂ : tendsto (λ x : ℝ, 1 + t / x) at_top (𝓝 1) :=
by simpa using (tendsto_inv_at_top_zero.const_mul t).const_add 1,
refine (eventually_ge_of_tendsto_gt h₁ h₂).mono... | lemma | tendsto_one_plus_div_rpow_exp | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"eventually_ge_of_tendsto_gt",
"exp",
"mul_comm"
] | The function `(1 + t/x) ^ x` tends to `exp t` at `+∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_one_plus_div_pow_exp (t : ℝ) :
tendsto (λ (x : ℕ), (1 + t / (x:ℝ)) ^ x) at_top (𝓝 (real.exp t)) | ((tendsto_one_plus_div_rpow_exp t).comp tendsto_coe_nat_at_top_at_top).congr (by simp) | lemma | tendsto_one_plus_div_pow_exp | analysis.special_functions.pow | src/analysis/special_functions/pow/deriv.lean | [
"analysis.special_functions.pow.continuity",
"analysis.special_functions.complex.log_deriv",
"analysis.calculus.extend_deriv",
"analysis.calculus.deriv.prod",
"analysis.special_functions.log.deriv",
"analysis.special_functions.trigonometric.deriv"
] | [
"real.exp",
"tendsto_coe_nat_at_top_at_top",
"tendsto_one_plus_div_rpow_exp"
] | The function `(1 + t/x) ^ x` tends to `exp t` at `+∞` for naturals `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rpow (x : ℝ≥0) (y : ℝ) : ℝ≥0 | ⟨(x : ℝ) ^ y, real.rpow_nonneg_of_nonneg x.2 y⟩ | def | nnreal.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.rpow_nonneg_of_nonneg"
] | The nonnegative real power function `x^y`, defined for `x : ℝ≥0` and `y : ℝ ` as the
restriction of the real power function. For `x > 0`, it is equal to `exp (y log x)`. For `x = 0`,
one sets `0 ^ 0 = 1` and `0 ^ y = 0` for `y ≠ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rpow_eq_pow (x : ℝ≥0) (y : ℝ) : rpow x y = x ^ y | rfl | lemma | nnreal.rpow_eq_pow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_rpow (x : ℝ≥0) (y : ℝ) : ((x ^ y : ℝ≥0) : ℝ) = (x : ℝ) ^ y | rfl | lemma | nnreal.coe_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_zero (x : ℝ≥0) : x ^ (0 : ℝ) = 1 | nnreal.eq $ real.rpow_zero _ | lemma | nnreal.rpow_zero | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.eq",
"real.rpow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_eq_zero_iff {x : ℝ≥0} {y : ℝ} : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 | begin
rw [← nnreal.coe_eq, coe_rpow, ← nnreal.coe_eq_zero],
exact real.rpow_eq_zero_iff_of_nonneg x.2
end | lemma | nnreal.rpow_eq_zero_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.coe_eq",
"nnreal.coe_eq_zero",
"real.rpow_eq_zero_iff_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_rpow {x : ℝ} (h : x ≠ 0) : (0 : ℝ≥0) ^ x = 0 | nnreal.eq $ real.zero_rpow h | lemma | nnreal.zero_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.eq",
"real.zero_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_one (x : ℝ≥0) : x ^ (1 : ℝ) = x | nnreal.eq $ real.rpow_one _ | lemma | nnreal.rpow_one | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.eq",
"real.rpow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_rpow (x : ℝ) : (1 : ℝ≥0) ^ x = 1 | nnreal.eq $ real.one_rpow _ | lemma | nnreal.one_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.eq",
"real.one_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_add {x : ℝ≥0} (hx : x ≠ 0) (y z : ℝ) : x ^ (y + z) = x ^ y * x ^ z | nnreal.eq $ real.rpow_add (pos_iff_ne_zero.2 hx) _ _ | lemma | nnreal.rpow_add | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.eq",
"real.rpow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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