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"categories": [
{"id": "derivatives", "name": "Derivatives", "icon": "∂"},
{"id": "integrals", "name": "Integrals", "icon": "∫"},
{"id": "limits", "name": "Limits", "icon": "lim"},
{"id": "series", "name": "Series", "icon": "Σ"}
],
"formulas": [
{
"id": "power_rule",
"category": "derivatives",
"name": "Power Rule",
"latex": "\\frac{d}{dx} x^3",
"tag": "derivative",
"params": {"variable": "x"},
"description": "Derivative of x³"
},
{
"id": "product_rule_ex",
"category": "derivatives",
"name": "Product Rule",
"latex": "\\frac{d}{dx} x \\sin(x)",
"tag": "derivative",
"params": {"variable": "x"},
"description": "Product rule example"
},
{
"id": "chain_rule_ex",
"category": "derivatives",
"name": "Chain Rule",
"latex": "\\frac{d}{dx} \\sin(x^2)",
"tag": "derivative",
"params": {"variable": "x"},
"description": "Chain rule example"
},
{
"id": "quotient_ex",
"category": "derivatives",
"name": "Quotient Rule",
"latex": "\\frac{d}{dx} \\frac{x}{\\cos(x)}",
"tag": "derivative",
"params": {"variable": "x"},
"description": "Quotient rule example"
},
{
"id": "exp_deriv",
"category": "derivatives",
"name": "Exponential",
"latex": "\\frac{d}{dx} e^{2x}",
"tag": "derivative",
"params": {"variable": "x"},
"description": "Derivative of e^(2x)"
},
{
"id": "log_deriv",
"category": "derivatives",
"name": "Logarithmic",
"latex": "\\frac{d}{dx} \\ln(x^2 + 1)",
"tag": "derivative",
"params": {"variable": "x"},
"description": "Derivative of ln(x²+1)"
},
{
"id": "second_deriv",
"category": "derivatives",
"name": "Second Derivative",
"latex": "\\frac{d^2}{dx^2} x^4",
"tag": "derivative",
"params": {"variable": "x", "order": 2},
"description": "Second derivative of x⁴"
},
{
"id": "poly_integral",
"category": "integrals",
"name": "Polynomial",
"latex": "\\int x^3 + 2x \\, dx",
"tag": "integral",
"params": {"variable": "x"},
"description": "Integral of x³ + 2x"
},
{
"id": "trig_integral",
"category": "integrals",
"name": "Trig Integral",
"latex": "\\int \\sin(x) \\, dx",
"tag": "integral",
"params": {"variable": "x"},
"description": "Integral of sin(x)"
},
{
"id": "exp_integral",
"category": "integrals",
"name": "Exponential",
"latex": "\\int e^{-x} \\, dx",
"tag": "integral",
"params": {"variable": "x"},
"description": "Integral of e^(-x)"
},
{
"id": "def_integral_x2",
"category": "integrals",
"name": "Definite ∫ x²",
"latex": "\\int_0^1 x^2 \\, dx",
"tag": "definite_integral",
"params": {"variable": "x", "lower": 0, "upper": 1},
"description": "∫₀¹ x² dx"
},
{
"id": "def_integral_sin",
"category": "integrals",
"name": "Definite ∫ sin",
"latex": "\\int_0^{\\pi} \\sin(x) \\, dx",
"tag": "definite_integral",
"params": {"variable": "x", "lower": 0, "upper": 3.14159},
"description": "∫₀^π sin(x) dx"
},
{
"id": "sinc_limit",
"category": "limits",
"name": "sin(x)/x",
"latex": "\\lim_{x \\to 0} \\frac{\\sin(x)}{x}",
"tag": "limit",
"params": {"variable": "x", "point": "0"},
"description": "Classic sinc limit"
},
{
"id": "inf_limit",
"category": "limits",
"name": "Limit at ∞",
"latex": "\\lim_{x \\to \\infty} \\frac{1}{x}",
"tag": "limit",
"params": {"variable": "x", "point": "oo"},
"description": "Limit as x → ∞"
},
{
"id": "exp_limit",
"category": "limits",
"name": "e definition",
"latex": "\\lim_{x \\to \\infty} (1 + \\frac{1}{x})^x",
"tag": "limit",
"params": {"variable": "x", "point": "oo"},
"description": "Limit definition of e"
},
{
"id": "lhopital_ex",
"category": "limits",
"name": "L'Hôpital",
"latex": "\\lim_{x \\to 0} \\frac{e^x - 1}{x}",
"tag": "limit",
"params": {"variable": "x", "point": "0"},
"description": "L'Hôpital's rule example"
},
{
"id": "taylor_sin",
"category": "series",
"name": "Taylor sin(x)",
"latex": "\\sin(x)",
"tag": "taylor",
"params": {"variable": "x", "point": 0, "order": 7},
"description": "Taylor series of sin(x)"
},
{
"id": "taylor_exp",
"category": "series",
"name": "Taylor eˣ",
"latex": "e^x",
"tag": "taylor",
"params": {"variable": "x", "point": 0, "order": 6},
"description": "Taylor series of eˣ"
},
{
"id": "taylor_cos",
"category": "series",
"name": "Taylor cos(x)",
"latex": "\\cos(x)",
"tag": "taylor",
"params": {"variable": "x", "point": 0, "order": 6},
"description": "Taylor series of cos(x)"
},
{
"id": "taylor_ln",
"category": "series",
"name": "Taylor ln(1+x)",
"latex": "\\ln(1+x)",
"tag": "taylor",
"params": {"variable": "x", "point": 0, "order": 6},
"description": "Taylor series of ln(1+x)"
}
]
} |