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
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
fderiv_within_inv (x_ne_zero : x ≠ 0) (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (λx, x⁻¹) s x = smul_right (1 : 𝕜 →L[𝕜] 𝕜) (-(x^2)⁻¹) | begin
rw differentiable_at.fderiv_within (differentiable_at_inv.2 x_ne_zero) hxs,
exact fderiv_inv
end | lemma | fderiv_within_inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"differentiable_at.fderiv_within",
"fderiv_inv",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.inv
(hc : has_deriv_within_at c c' s x) (hx : c x ≠ 0) :
has_deriv_within_at (λ y, (c y)⁻¹) (- c' / (c x)^2) s x | begin
convert (has_deriv_at_inv hx).comp_has_deriv_within_at x hc,
field_simp
end | lemma | has_deriv_within_at.inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"has_deriv_at_inv",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.inv (hc : has_deriv_at c c' x) (hx : c x ≠ 0) :
has_deriv_at (λ y, (c y)⁻¹) (- c' / (c x)^2) x | begin
rw ← has_deriv_within_at_univ at *,
exact hc.inv hx
end | lemma | has_deriv_at.inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"has_deriv_at",
"has_deriv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.inv (hf : differentiable_within_at 𝕜 h S z) (hz : h z ≠ 0) :
differentiable_within_at 𝕜 (λx, (h x)⁻¹) S z | (differentiable_at_inv.mpr hz).comp_differentiable_within_at z hf | lemma | differentiable_within_at.inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.inv (hf : differentiable_at 𝕜 h z) (hz : h z ≠ 0) :
differentiable_at 𝕜 (λx, (h x)⁻¹) z | (differentiable_at_inv.mpr hz).comp z hf | lemma | differentiable_at.inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.inv (hf : differentiable_on 𝕜 h S) (hz : ∀ x ∈ S, h x ≠ 0) :
differentiable_on 𝕜 (λx, (h x)⁻¹) S | λx h, (hf x h).inv (hz x h) | lemma | differentiable_on.inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.inv (hf : differentiable 𝕜 h) (hz : ∀ x, h x ≠ 0) :
differentiable 𝕜 (λx, (h x)⁻¹) | λx, (hf x).inv (hz x) | lemma | differentiable.inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_inv' (hc : differentiable_within_at 𝕜 c s x) (hx : c x ≠ 0)
(hxs : unique_diff_within_at 𝕜 s x) :
deriv_within (λx, (c x)⁻¹) s x = - (deriv_within c s x) / (c x)^2 | (hc.has_deriv_within_at.inv hx).deriv_within hxs | lemma | deriv_within_inv' | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_inv'' (hc : differentiable_at 𝕜 c x) (hx : c x ≠ 0) :
deriv (λx, (c x)⁻¹) x = - (deriv c x) / (c x)^2 | (hc.has_deriv_at.inv hx).deriv | lemma | deriv_inv'' | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.div
(hc : has_deriv_within_at c c' s x) (hd : has_deriv_within_at d d' s x) (hx : d x ≠ 0) :
has_deriv_within_at (λ y, c y / d y) ((c' * d x - c x * d') / (d x)^2) s x | begin
convert hc.mul ((has_deriv_at_inv hx).comp_has_deriv_within_at x hd),
{ simp only [div_eq_mul_inv] },
{ field_simp, ring }
end | lemma | has_deriv_within_at.div | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"div_eq_mul_inv",
"has_deriv_at_inv",
"has_deriv_within_at",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.div (hc : has_strict_deriv_at c c' x) (hd : has_strict_deriv_at d d' x)
(hx : d x ≠ 0) :
has_strict_deriv_at (λ y, c y / d y) ((c' * d x - c x * d') / (d x)^2) x | begin
convert hc.mul ((has_strict_deriv_at_inv hx).comp x hd),
{ simp only [div_eq_mul_inv] },
{ field_simp, ring }
end | lemma | has_strict_deriv_at.div | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"div_eq_mul_inv",
"has_strict_deriv_at",
"has_strict_deriv_at_inv",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.div (hc : has_deriv_at c c' x) (hd : has_deriv_at d d' x) (hx : d x ≠ 0) :
has_deriv_at (λ y, c y / d y) ((c' * d x - c x * d') / (d x)^2) x | begin
rw ← has_deriv_within_at_univ at *,
exact hc.div hd hx
end | lemma | has_deriv_at.div | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"has_deriv_at",
"has_deriv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.div
(hc : differentiable_within_at 𝕜 c s x) (hd : differentiable_within_at 𝕜 d s x) (hx : d x ≠ 0) :
differentiable_within_at 𝕜 (λx, c x / d x) s x | ((hc.has_deriv_within_at).div (hd.has_deriv_within_at) hx).differentiable_within_at | lemma | differentiable_within_at.div | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.div
(hc : differentiable_at 𝕜 c x) (hd : differentiable_at 𝕜 d x) (hx : d x ≠ 0) :
differentiable_at 𝕜 (λx, c x / d x) x | ((hc.has_deriv_at).div (hd.has_deriv_at) hx).differentiable_at | lemma | differentiable_at.div | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.div
(hc : differentiable_on 𝕜 c s) (hd : differentiable_on 𝕜 d s) (hx : ∀ x ∈ s, d x ≠ 0) :
differentiable_on 𝕜 (λx, c x / d x) s | λx h, (hc x h).div (hd x h) (hx x h) | lemma | differentiable_on.div | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.div
(hc : differentiable 𝕜 c) (hd : differentiable 𝕜 d) (hx : ∀ x, d x ≠ 0) :
differentiable 𝕜 (λx, c x / d x) | λx, (hc x).div (hd x) (hx x) | lemma | differentiable.div | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_div
(hc : differentiable_within_at 𝕜 c s x) (hd : differentiable_within_at 𝕜 d s x) (hx : d x ≠ 0)
(hxs : unique_diff_within_at 𝕜 s x) :
deriv_within (λx, c x / d x) s x
= ((deriv_within c s x) * d x - c x * (deriv_within d s x)) / (d x)^2 | ((hc.has_deriv_within_at).div (hd.has_deriv_within_at) hx).deriv_within hxs | lemma | deriv_within_div | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_div
(hc : differentiable_at 𝕜 c x) (hd : differentiable_at 𝕜 d x) (hx : d x ≠ 0) :
deriv (λx, c x / d x) x = ((deriv c x) * d x - c x * (deriv d x)) / (d x)^2 | ((hc.has_deriv_at).div (hd.has_deriv_at) hx).deriv | lemma | deriv_div | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.has_strict_fderiv_at_equiv {f : 𝕜 → 𝕜} {f' x : 𝕜}
(hf : has_strict_deriv_at f f' x) (hf' : f' ≠ 0) :
has_strict_fderiv_at f
(continuous_linear_equiv.units_equiv_aut 𝕜 (units.mk0 f' hf') : 𝕜 →L[𝕜] 𝕜) x | hf | theorem | has_strict_deriv_at.has_strict_fderiv_at_equiv | analysis.calculus.deriv | src/analysis/calculus/deriv/inverse.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.fderiv.equiv"
] | [
"continuous_linear_equiv.units_equiv_aut",
"has_strict_deriv_at",
"has_strict_fderiv_at",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.has_fderiv_at_equiv {f : 𝕜 → 𝕜} {f' x : 𝕜} (hf : has_deriv_at f f' x)
(hf' : f' ≠ 0) :
has_fderiv_at f (continuous_linear_equiv.units_equiv_aut 𝕜 (units.mk0 f' hf') : 𝕜 →L[𝕜] 𝕜) x | hf | theorem | has_deriv_at.has_fderiv_at_equiv | analysis.calculus.deriv | src/analysis/calculus/deriv/inverse.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.fderiv.equiv"
] | [
"continuous_linear_equiv.units_equiv_aut",
"has_deriv_at",
"has_fderiv_at",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.of_local_left_inverse {f g : 𝕜 → 𝕜} {f' a : 𝕜}
(hg : continuous_at g a) (hf : has_strict_deriv_at f f' (g a)) (hf' : f' ≠ 0)
(hfg : ∀ᶠ y in 𝓝 a, f (g y) = y) :
has_strict_deriv_at g f'⁻¹ a | (hf.has_strict_fderiv_at_equiv hf').of_local_left_inverse hg hfg | theorem | has_strict_deriv_at.of_local_left_inverse | analysis.calculus.deriv | src/analysis/calculus/deriv/inverse.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.fderiv.equiv"
] | [
"continuous_at",
"has_strict_deriv_at"
] | If `f (g y) = y` for `y` in some neighborhood of `a`, `g` is continuous at `a`, and `f` has an
invertible derivative `f'` at `g a` in the strict sense, then `g` has the derivative `f'⁻¹` at `a`
in the strict sense.
This is one of the easy parts of the inverse function theorem: it assumes that we already have an
invers... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_homeomorph.has_strict_deriv_at_symm (f : local_homeomorph 𝕜 𝕜) {a f' : 𝕜}
(ha : a ∈ f.target) (hf' : f' ≠ 0) (htff' : has_strict_deriv_at f f' (f.symm a)) :
has_strict_deriv_at f.symm f'⁻¹ a | htff'.of_local_left_inverse (f.symm.continuous_at ha) hf' (f.eventually_right_inverse ha) | lemma | local_homeomorph.has_strict_deriv_at_symm | analysis.calculus.deriv | src/analysis/calculus/deriv/inverse.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.fderiv.equiv"
] | [
"has_strict_deriv_at",
"local_homeomorph"
] | If `f` is a local homeomorphism defined on a neighbourhood of `f.symm a`, and `f` has a
nonzero derivative `f'` at `f.symm a` in the strict sense, then `f.symm` has the derivative `f'⁻¹`
at `a` in the strict sense.
This is one of the easy parts of the inverse function theorem: it assumes that we already have
an invers... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at.of_local_left_inverse {f g : 𝕜 → 𝕜} {f' a : 𝕜}
(hg : continuous_at g a) (hf : has_deriv_at f f' (g a)) (hf' : f' ≠ 0)
(hfg : ∀ᶠ y in 𝓝 a, f (g y) = y) :
has_deriv_at g f'⁻¹ a | (hf.has_fderiv_at_equiv hf').of_local_left_inverse hg hfg | theorem | has_deriv_at.of_local_left_inverse | analysis.calculus.deriv | src/analysis/calculus/deriv/inverse.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.fderiv.equiv"
] | [
"continuous_at",
"has_deriv_at"
] | If `f (g y) = y` for `y` in some neighborhood of `a`, `g` is continuous at `a`, and `f` has an
invertible derivative `f'` at `g a`, then `g` has the derivative `f'⁻¹` at `a`.
This is one of the easy parts of the inverse function theorem: it assumes that we already have
an inverse function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_homeomorph.has_deriv_at_symm (f : local_homeomorph 𝕜 𝕜) {a f' : 𝕜}
(ha : a ∈ f.target) (hf' : f' ≠ 0) (htff' : has_deriv_at f f' (f.symm a)) :
has_deriv_at f.symm f'⁻¹ a | htff'.of_local_left_inverse (f.symm.continuous_at ha) hf' (f.eventually_right_inverse ha) | lemma | local_homeomorph.has_deriv_at_symm | analysis.calculus.deriv | src/analysis/calculus/deriv/inverse.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.fderiv.equiv"
] | [
"has_deriv_at",
"local_homeomorph"
] | If `f` is a local homeomorphism defined on a neighbourhood of `f.symm a`, and `f` has an
nonzero derivative `f'` at `f.symm a`, then `f.symm` has the derivative `f'⁻¹` at `a`.
This is one of the easy parts of the inverse function theorem: it assumes that we already have
an inverse function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at.eventually_ne (h : has_deriv_at f f' x) (hf' : f' ≠ 0) :
∀ᶠ z in 𝓝[≠] x, f z ≠ f x | (has_deriv_at_iff_has_fderiv_at.1 h).eventually_ne
⟨‖f'‖⁻¹, λ z, by field_simp [norm_smul, mt norm_eq_zero.1 hf']⟩ | lemma | has_deriv_at.eventually_ne | analysis.calculus.deriv | src/analysis/calculus/deriv/inverse.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.fderiv.equiv"
] | [
"has_deriv_at",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.tendsto_punctured_nhds (h : has_deriv_at f f' x) (hf' : f' ≠ 0) :
tendsto f (𝓝[≠] x) (𝓝[≠] (f x)) | tendsto_nhds_within_of_tendsto_nhds_of_eventually_within _
h.continuous_at.continuous_within_at (h.eventually_ne hf') | lemma | has_deriv_at.tendsto_punctured_nhds | analysis.calculus.deriv | src/analysis/calculus/deriv/inverse.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.fderiv.equiv"
] | [
"has_deriv_at",
"tendsto_nhds_within_of_tendsto_nhds_of_eventually_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_differentiable_within_at_of_local_left_inverse_has_deriv_within_at_zero
{f g : 𝕜 → 𝕜} {a : 𝕜} {s t : set 𝕜} (ha : a ∈ s) (hsu : unique_diff_within_at 𝕜 s a)
(hf : has_deriv_within_at f 0 t (g a)) (hst : maps_to g s t) (hfg : f ∘ g =ᶠ[𝓝[s] a] id) :
¬differentiable_within_at 𝕜 g s a | begin
intro hg,
have := (hf.comp a hg.has_deriv_within_at hst).congr_of_eventually_eq_of_mem hfg.symm ha,
simpa using hsu.eq_deriv _ this (has_deriv_within_at_id _ _)
end | theorem | not_differentiable_within_at_of_local_left_inverse_has_deriv_within_at_zero | analysis.calculus.deriv | src/analysis/calculus/deriv/inverse.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.fderiv.equiv"
] | [
"differentiable_within_at",
"has_deriv_within_at",
"has_deriv_within_at_id",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_differentiable_at_of_local_left_inverse_has_deriv_at_zero
{f g : 𝕜 → 𝕜} {a : 𝕜} (hf : has_deriv_at f 0 (g a)) (hfg : f ∘ g =ᶠ[𝓝 a] id) :
¬differentiable_at 𝕜 g a | begin
intro hg,
have := (hf.comp a hg.has_deriv_at).congr_of_eventually_eq hfg.symm,
simpa using this.unique (has_deriv_at_id a)
end | theorem | not_differentiable_at_of_local_left_inverse_has_deriv_at_zero | analysis.calculus.deriv | src/analysis/calculus/deriv/inverse.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.fderiv.equiv"
] | [
"differentiable_at",
"has_deriv_at",
"has_deriv_at_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.has_deriv_at_filter : has_deriv_at_filter e (e 1) x L | e.has_fderiv_at_filter.has_deriv_at_filter | lemma | continuous_linear_map.has_deriv_at_filter | analysis.calculus.deriv | src/analysis/calculus/deriv/linear.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.linear"
] | [
"has_deriv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.has_strict_deriv_at : has_strict_deriv_at e (e 1) x | e.has_strict_fderiv_at.has_strict_deriv_at | lemma | continuous_linear_map.has_strict_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/linear.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.linear"
] | [
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.has_deriv_at : has_deriv_at e (e 1) x | e.has_deriv_at_filter | lemma | continuous_linear_map.has_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/linear.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.linear"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.has_deriv_within_at : has_deriv_within_at e (e 1) s x | e.has_deriv_at_filter | lemma | continuous_linear_map.has_deriv_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/linear.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.linear"
] | [
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.deriv : deriv e x = e 1 | e.has_deriv_at.deriv | lemma | continuous_linear_map.deriv | analysis.calculus.deriv | src/analysis/calculus/deriv/linear.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.linear"
] | [
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.deriv_within (hxs : unique_diff_within_at 𝕜 s x) :
deriv_within e s x = e 1 | e.has_deriv_within_at.deriv_within hxs | lemma | continuous_linear_map.deriv_within | analysis.calculus.deriv | src/analysis/calculus/deriv/linear.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.linear"
] | [
"deriv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.has_deriv_at_filter : has_deriv_at_filter e (e 1) x L | e.to_continuous_linear_map₁.has_deriv_at_filter | lemma | linear_map.has_deriv_at_filter | analysis.calculus.deriv | src/analysis/calculus/deriv/linear.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.linear"
] | [
"has_deriv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.has_strict_deriv_at : has_strict_deriv_at e (e 1) x | e.to_continuous_linear_map₁.has_strict_deriv_at | lemma | linear_map.has_strict_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/linear.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.linear"
] | [
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.has_deriv_at : has_deriv_at e (e 1) x | e.has_deriv_at_filter | lemma | linear_map.has_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/linear.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.linear"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.has_deriv_within_at : has_deriv_within_at e (e 1) s x | e.has_deriv_at_filter | lemma | linear_map.has_deriv_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/linear.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.linear"
] | [
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.deriv : deriv e x = e 1 | e.has_deriv_at.deriv | lemma | linear_map.deriv | analysis.calculus.deriv | src/analysis/calculus/deriv/linear.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.linear"
] | [
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.deriv_within (hxs : unique_diff_within_at 𝕜 s x) :
deriv_within e s x = e 1 | e.has_deriv_within_at.deriv_within hxs | lemma | linear_map.deriv_within | analysis.calculus.deriv | src/analysis/calculus/deriv/linear.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.linear"
] | [
"deriv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.smul
(hc : has_deriv_within_at c c' s x) (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ y, c y • f y) (c x • f' + c' • f x) s x | by simpa using (has_fderiv_within_at.smul hc hf).has_deriv_within_at | theorem | has_deriv_within_at.smul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_within_at",
"has_fderiv_within_at.smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.smul
(hc : has_deriv_at c c' x) (hf : has_deriv_at f f' x) :
has_deriv_at (λ y, c y • f y) (c x • f' + c' • f x) x | begin
rw [← has_deriv_within_at_univ] at *,
exact hc.smul hf
end | theorem | has_deriv_at.smul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at",
"has_deriv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.smul
(hc : has_strict_deriv_at c c' x) (hf : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ y, c y • f y) (c x • f' + c' • f x) x | by simpa using (hc.smul hf).has_strict_deriv_at | theorem | has_strict_deriv_at.smul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_smul (hxs : unique_diff_within_at 𝕜 s x)
(hc : differentiable_within_at 𝕜 c s x) (hf : differentiable_within_at 𝕜 f s x) :
deriv_within (λ y, c y • f y) s x = c x • deriv_within f s x + (deriv_within c s x) • f x | (hc.has_deriv_within_at.smul hf.has_deriv_within_at).deriv_within hxs | lemma | deriv_within_smul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_smul (hc : differentiable_at 𝕜 c x) (hf : differentiable_at 𝕜 f x) :
deriv (λ y, c y • f y) x = c x • deriv f x + (deriv c x) • f x | (hc.has_deriv_at.smul hf.has_deriv_at).deriv | lemma | deriv_smul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.smul_const
(hc : has_strict_deriv_at c c' x) (f : F) :
has_strict_deriv_at (λ y, c y • f) (c' • f) x | begin
have := hc.smul (has_strict_deriv_at_const x f),
rwa [smul_zero, zero_add] at this,
end | theorem | has_strict_deriv_at.smul_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_strict_deriv_at",
"has_strict_deriv_at_const",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.smul_const
(hc : has_deriv_within_at c c' s x) (f : F) :
has_deriv_within_at (λ y, c y • f) (c' • f) s x | begin
have := hc.smul (has_deriv_within_at_const x s f),
rwa [smul_zero, zero_add] at this
end | theorem | has_deriv_within_at.smul_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_within_at",
"has_deriv_within_at_const",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.smul_const
(hc : has_deriv_at c c' x) (f : F) :
has_deriv_at (λ y, c y • f) (c' • f) x | begin
rw [← has_deriv_within_at_univ] at *,
exact hc.smul_const f
end | theorem | has_deriv_at.smul_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at",
"has_deriv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_smul_const (hxs : unique_diff_within_at 𝕜 s x)
(hc : differentiable_within_at 𝕜 c s x) (f : F) :
deriv_within (λ y, c y • f) s x = (deriv_within c s x) • f | (hc.has_deriv_within_at.smul_const f).deriv_within hxs | lemma | deriv_within_smul_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_smul_const (hc : differentiable_at 𝕜 c x) (f : F) :
deriv (λ y, c y • f) x = (deriv c x) • f | (hc.has_deriv_at.smul_const f).deriv | lemma | deriv_smul_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.const_smul
(c : R) (hf : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ y, c • f y) (c • f') x | by simpa using (hf.const_smul c).has_strict_deriv_at | theorem | has_strict_deriv_at.const_smul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter.const_smul
(c : R) (hf : has_deriv_at_filter f f' x L) :
has_deriv_at_filter (λ y, c • f y) (c • f') x L | by simpa using (hf.const_smul c).has_deriv_at_filter | theorem | has_deriv_at_filter.const_smul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.const_smul
(c : R) (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ y, c • f y) (c • f') s x | hf.const_smul c | theorem | has_deriv_within_at.const_smul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.const_smul (c : R) (hf : has_deriv_at f f' x) :
has_deriv_at (λ y, c • f y) (c • f') x | hf.const_smul c | theorem | has_deriv_at.const_smul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_const_smul (hxs : unique_diff_within_at 𝕜 s x)
(c : R) (hf : differentiable_within_at 𝕜 f s x) :
deriv_within (λ y, c • f y) s x = c • deriv_within f s x | (hf.has_deriv_within_at.const_smul c).deriv_within hxs | lemma | deriv_within_const_smul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_const_smul (c : R) (hf : differentiable_at 𝕜 f x) :
deriv (λ y, c • f y) x = c • deriv f x | (hf.has_deriv_at.const_smul c).deriv | lemma | deriv_const_smul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.mul
(hc : has_deriv_within_at c c' s x) (hd : has_deriv_within_at d d' s x) :
has_deriv_within_at (λ y, c y * d y) (c' * d x + c x * d') s x | begin
have := (has_fderiv_within_at.mul' hc hd).has_deriv_within_at,
rwa [continuous_linear_map.add_apply, continuous_linear_map.smul_apply,
continuous_linear_map.smul_right_apply, continuous_linear_map.smul_right_apply,
continuous_linear_map.smul_right_apply, continuous_linear_map.one_apply,
one_... | theorem | has_deriv_within_at.mul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"continuous_linear_map.add_apply",
"continuous_linear_map.one_apply",
"continuous_linear_map.smul_apply",
"continuous_linear_map.smul_right_apply",
"has_deriv_within_at",
"has_fderiv_within_at.mul'",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.mul (hc : has_deriv_at c c' x) (hd : has_deriv_at d d' x) :
has_deriv_at (λ y, c y * d y) (c' * d x + c x * d') x | begin
rw [← has_deriv_within_at_univ] at *,
exact hc.mul hd
end | theorem | has_deriv_at.mul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at",
"has_deriv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.mul
(hc : has_strict_deriv_at c c' x) (hd : has_strict_deriv_at d d' x) :
has_strict_deriv_at (λ y, c y * d y) (c' * d x + c x * d') x | begin
have := (has_strict_fderiv_at.mul' hc hd).has_strict_deriv_at,
rwa [continuous_linear_map.add_apply, continuous_linear_map.smul_apply,
continuous_linear_map.smul_right_apply, continuous_linear_map.smul_right_apply,
continuous_linear_map.smul_right_apply, continuous_linear_map.one_apply,
one_... | theorem | has_strict_deriv_at.mul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"continuous_linear_map.add_apply",
"continuous_linear_map.one_apply",
"continuous_linear_map.smul_apply",
"continuous_linear_map.smul_right_apply",
"has_strict_deriv_at",
"has_strict_fderiv_at.mul'",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_mul (hxs : unique_diff_within_at 𝕜 s x)
(hc : differentiable_within_at 𝕜 c s x) (hd : differentiable_within_at 𝕜 d s x) :
deriv_within (λ y, c y * d y) s x = deriv_within c s x * d x + c x * deriv_within d s x | (hc.has_deriv_within_at.mul hd.has_deriv_within_at).deriv_within hxs | lemma | deriv_within_mul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_mul (hc : differentiable_at 𝕜 c x) (hd : differentiable_at 𝕜 d x) :
deriv (λ y, c y * d y) x = deriv c x * d x + c x * deriv d x | (hc.has_deriv_at.mul hd.has_deriv_at).deriv | lemma | deriv_mul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.mul_const (hc : has_deriv_within_at c c' s x) (d : 𝔸) :
has_deriv_within_at (λ y, c y * d) (c' * d) s x | begin
convert hc.mul (has_deriv_within_at_const x s d),
rw [mul_zero, add_zero]
end | theorem | has_deriv_within_at.mul_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_within_at",
"has_deriv_within_at_const",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.mul_const (hc : has_deriv_at c c' x) (d : 𝔸) :
has_deriv_at (λ y, c y * d) (c' * d) x | begin
rw [← has_deriv_within_at_univ] at *,
exact hc.mul_const d
end | theorem | has_deriv_at.mul_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at",
"has_deriv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_mul_const (c : 𝕜) : has_deriv_at (λ x, x * c) c x | by simpa only [one_mul] using (has_deriv_at_id' x).mul_const c | theorem | has_deriv_at_mul_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at",
"has_deriv_at_id'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.mul_const (hc : has_strict_deriv_at c c' x) (d : 𝔸) :
has_strict_deriv_at (λ y, c y * d) (c' * d) x | begin
convert hc.mul (has_strict_deriv_at_const x d),
rw [mul_zero, add_zero]
end | theorem | has_strict_deriv_at.mul_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_strict_deriv_at",
"has_strict_deriv_at_const",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_mul_const (hxs : unique_diff_within_at 𝕜 s x)
(hc : differentiable_within_at 𝕜 c s x) (d : 𝔸) :
deriv_within (λ y, c y * d) s x = deriv_within c s x * d | (hc.has_deriv_within_at.mul_const d).deriv_within hxs | lemma | deriv_within_mul_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_mul_const (hc : differentiable_at 𝕜 c x) (d : 𝔸) :
deriv (λ y, c y * d) x = deriv c x * d | (hc.has_deriv_at.mul_const d).deriv | lemma | deriv_mul_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_mul_const_field (v : 𝕜') :
deriv (λ y, u y * v) x = deriv u x * v | begin
by_cases hu : differentiable_at 𝕜 u x,
{ exact deriv_mul_const hu v },
{ rw [deriv_zero_of_not_differentiable_at hu, zero_mul],
rcases eq_or_ne v 0 with rfl|hd,
{ simp only [mul_zero, deriv_const] },
{ refine deriv_zero_of_not_differentiable_at (mt (λ H, _) hu),
simpa only [mul_inv_cancel... | lemma | deriv_mul_const_field | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"deriv_const",
"deriv_mul_const",
"deriv_zero_of_not_differentiable_at",
"differentiable_at",
"eq_or_ne",
"mul_inv_cancel_right₀",
"mul_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_mul_const_field' (v : 𝕜') : deriv (λ x, u x * v) = λ x, deriv u x * v | funext $ λ _, deriv_mul_const_field v | lemma | deriv_mul_const_field' | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"deriv_mul_const_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.const_mul (c : 𝔸) (hd : has_deriv_within_at d d' s x) :
has_deriv_within_at (λ y, c * d y) (c * d') s x | begin
convert (has_deriv_within_at_const x s c).mul hd,
rw [zero_mul, zero_add]
end | theorem | has_deriv_within_at.const_mul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_within_at",
"has_deriv_within_at_const",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.const_mul (c : 𝔸) (hd : has_deriv_at d d' x) :
has_deriv_at (λ y, c * d y) (c * d') x | begin
rw [← has_deriv_within_at_univ] at *,
exact hd.const_mul c
end | theorem | has_deriv_at.const_mul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at",
"has_deriv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.const_mul (c : 𝔸) (hd : has_strict_deriv_at d d' x) :
has_strict_deriv_at (λ y, c * d y) (c * d') x | begin
convert (has_strict_deriv_at_const _ _).mul hd,
rw [zero_mul, zero_add]
end | theorem | has_strict_deriv_at.const_mul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_strict_deriv_at",
"has_strict_deriv_at_const",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_const_mul (hxs : unique_diff_within_at 𝕜 s x)
(c : 𝔸) (hd : differentiable_within_at 𝕜 d s x) :
deriv_within (λ y, c * d y) s x = c * deriv_within d s x | (hd.has_deriv_within_at.const_mul c).deriv_within hxs | lemma | deriv_within_const_mul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_const_mul (c : 𝔸) (hd : differentiable_at 𝕜 d x) :
deriv (λ y, c * d y) x = c * deriv d x | (hd.has_deriv_at.const_mul c).deriv | lemma | deriv_const_mul | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_const_mul_field (u : 𝕜') : deriv (λ y, u * v y) x = u * deriv v x | by simp only [mul_comm u, deriv_mul_const_field] | lemma | deriv_const_mul_field | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"deriv_mul_const_field",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_const_mul_field' (u : 𝕜') : deriv (λ x, u * v x) = λ x, u * deriv v x | funext (λ x, deriv_const_mul_field u) | lemma | deriv_const_mul_field' | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"deriv_const_mul_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.div_const (hc : has_deriv_at c c' x) (d : 𝕜') :
has_deriv_at (λ x, c x / d) (c' / d) x | by simpa only [div_eq_mul_inv] using hc.mul_const d⁻¹ | lemma | has_deriv_at.div_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"div_eq_mul_inv",
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.div_const (hc : has_deriv_within_at c c' s x) (d : 𝕜') :
has_deriv_within_at (λ x, c x / d) (c' / d) s x | by simpa only [div_eq_mul_inv] using hc.mul_const d⁻¹ | lemma | has_deriv_within_at.div_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"div_eq_mul_inv",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.div_const (hc : has_strict_deriv_at c c' x) (d : 𝕜') :
has_strict_deriv_at (λ x, c x / d) (c' / d) x | by simpa only [div_eq_mul_inv] using hc.mul_const d⁻¹ | lemma | has_strict_deriv_at.div_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"div_eq_mul_inv",
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.div_const (hc : differentiable_within_at 𝕜 c s x) (d : 𝕜') :
differentiable_within_at 𝕜 (λx, c x / d) s x | (hc.has_deriv_within_at.div_const _).differentiable_within_at | lemma | differentiable_within_at.div_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.div_const (hc : differentiable_at 𝕜 c x) (d : 𝕜') :
differentiable_at 𝕜 (λ x, c x / d) x | (hc.has_deriv_at.div_const _).differentiable_at | lemma | differentiable_at.div_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.div_const (hc : differentiable_on 𝕜 c s) (d : 𝕜') :
differentiable_on 𝕜 (λx, c x / d) s | λ x hx, (hc x hx).div_const d | lemma | differentiable_on.div_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.div_const (hc : differentiable 𝕜 c) (d : 𝕜') :
differentiable 𝕜 (λx, c x / d) | λ x, (hc x).div_const d | lemma | differentiable.div_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_div_const (hc : differentiable_within_at 𝕜 c s x) (d : 𝕜')
(hxs : unique_diff_within_at 𝕜 s x) :
deriv_within (λx, c x / d) s x = (deriv_within c s x) / d | by simp [div_eq_inv_mul, deriv_within_const_mul, hc, hxs] | lemma | deriv_within_div_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv_within",
"deriv_within_const_mul",
"differentiable_within_at",
"div_eq_inv_mul",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_div_const (d : 𝕜') :
deriv (λx, c x / d) x = (deriv c x) / d | by simp only [div_eq_mul_inv, deriv_mul_const_field] | lemma | deriv_div_const | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"deriv_mul_const_field",
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.clm_comp (hc : has_strict_deriv_at c c' x)
(hd : has_strict_deriv_at d d' x) :
has_strict_deriv_at (λ y, (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') x | begin
have := (hc.has_strict_fderiv_at.clm_comp hd.has_strict_fderiv_at).has_strict_deriv_at,
rwa [add_apply, comp_apply, comp_apply, smul_right_apply, smul_right_apply, one_apply, one_smul,
one_smul, add_comm] at this,
end | lemma | has_strict_deriv_at.clm_comp | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_strict_deriv_at",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.clm_comp (hc : has_deriv_within_at c c' s x)
(hd : has_deriv_within_at d d' s x) :
has_deriv_within_at (λ y, (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') s x | begin
have := (hc.has_fderiv_within_at.clm_comp hd.has_fderiv_within_at).has_deriv_within_at,
rwa [add_apply, comp_apply, comp_apply, smul_right_apply, smul_right_apply, one_apply, one_smul,
one_smul, add_comm] at this,
end | lemma | has_deriv_within_at.clm_comp | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_within_at",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.clm_comp (hc : has_deriv_at c c' x) (hd : has_deriv_at d d' x) :
has_deriv_at (λ y, (c y).comp (d y))
(c'.comp (d x) + (c x).comp d') x | begin
rw [← has_deriv_within_at_univ] at *,
exact hc.clm_comp hd
end | lemma | has_deriv_at.clm_comp | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at",
"has_deriv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_clm_comp (hc : differentiable_within_at 𝕜 c s x)
(hd : differentiable_within_at 𝕜 d s x) (hxs : unique_diff_within_at 𝕜 s x):
deriv_within (λ y, (c y).comp (d y)) s x =
((deriv_within c s x).comp (d x) + (c x).comp (deriv_within d s x)) | (hc.has_deriv_within_at.clm_comp hd.has_deriv_within_at).deriv_within hxs | lemma | deriv_within_clm_comp | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_clm_comp (hc : differentiable_at 𝕜 c x) (hd : differentiable_at 𝕜 d x) :
deriv (λ y, (c y).comp (d y)) x =
((deriv c x).comp (d x) + (c x).comp (deriv d x)) | (hc.has_deriv_at.clm_comp hd.has_deriv_at).deriv | lemma | deriv_clm_comp | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.clm_apply (hc : has_strict_deriv_at c c' x)
(hu : has_strict_deriv_at u u' x) :
has_strict_deriv_at (λ y, (c y) (u y)) (c' (u x) + c x u') x | begin
have := (hc.has_strict_fderiv_at.clm_apply hu.has_strict_fderiv_at).has_strict_deriv_at,
rwa [add_apply, comp_apply, flip_apply, smul_right_apply, smul_right_apply, one_apply, one_smul,
one_smul, add_comm] at this,
end | lemma | has_strict_deriv_at.clm_apply | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_strict_deriv_at",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.clm_apply (hc : has_deriv_within_at c c' s x)
(hu : has_deriv_within_at u u' s x) :
has_deriv_within_at (λ y, (c y) (u y)) (c' (u x) + c x u') s x | begin
have := (hc.has_fderiv_within_at.clm_apply hu.has_fderiv_within_at).has_deriv_within_at,
rwa [add_apply, comp_apply, flip_apply, smul_right_apply, smul_right_apply, one_apply, one_smul,
one_smul, add_comm] at this,
end | lemma | has_deriv_within_at.clm_apply | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_within_at",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.clm_apply (hc : has_deriv_at c c' x) (hu : has_deriv_at u u' x) :
has_deriv_at (λ y, (c y) (u y)) (c' (u x) + c x u') x | begin
have := (hc.has_fderiv_at.clm_apply hu.has_fderiv_at).has_deriv_at,
rwa [add_apply, comp_apply, flip_apply, smul_right_apply, smul_right_apply, one_apply, one_smul,
one_smul, add_comm] at this,
end | lemma | has_deriv_at.clm_apply | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_clm_apply (hxs : unique_diff_within_at 𝕜 s x)
(hc : differentiable_within_at 𝕜 c s x) (hu : differentiable_within_at 𝕜 u s x) :
deriv_within (λ y, (c y) (u y)) s x = (deriv_within c s x (u x) + c x (deriv_within u s x)) | (hc.has_deriv_within_at.clm_apply hu.has_deriv_within_at).deriv_within hxs | lemma | deriv_within_clm_apply | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_clm_apply (hc : differentiable_at 𝕜 c x) (hu : differentiable_at 𝕜 u x) :
deriv (λ y, (c y) (u y)) x = (deriv c x (u x) + c x (deriv u x)) | (hc.has_deriv_at.clm_apply hu.has_deriv_at).deriv | lemma | deriv_clm_apply | analysis.calculus.deriv | src/analysis/calculus/deriv/mul.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at (x : 𝕜) :
has_strict_deriv_at (λx, p.eval x) (p.derivative.eval x) x | begin
induction p using polynomial.induction_on',
case h_add : p q hp hq { simpa using hp.add hq },
case h_monomial : n a { simpa [mul_assoc] using (has_strict_deriv_at_pow n x).const_mul a }
end | lemma | polynomial.has_strict_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"has_strict_deriv_at",
"has_strict_deriv_at_pow",
"mul_assoc",
"polynomial.induction_on'"
] | The derivative (in the analysis sense) of a polynomial `p` is given by `p.derivative`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_deriv_at_aeval (x : 𝕜) :
has_strict_deriv_at (λx, aeval x q) (aeval x q.derivative) x | by simpa only [aeval_def, eval₂_eq_eval_map, derivative_map]
using (q.map (algebra_map R 𝕜)).has_strict_deriv_at x | lemma | polynomial.has_strict_deriv_at_aeval | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"algebra_map",
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at (x : 𝕜) : has_deriv_at (λx, p.eval x) (p.derivative.eval x) x | (p.has_strict_deriv_at x).has_deriv_at | lemma | polynomial.has_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"has_deriv_at"
] | The derivative (in the analysis sense) of a polynomial `p` is given by `p.derivative`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_aeval (x : 𝕜) :
has_deriv_at (λx, aeval x q) (aeval x q.derivative) x | (q.has_strict_deriv_at_aeval x).has_deriv_at | lemma | polynomial.has_deriv_at_aeval | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at (x : 𝕜) (s : set 𝕜) :
has_deriv_within_at (λx, p.eval x) (p.derivative.eval x) s x | (p.has_deriv_at x).has_deriv_within_at | theorem | polynomial.has_deriv_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.