Standard Differentials and Integrals

A comprehensive reference table of common differential and integral formulas in calculus, including polynomial, logarithmic, exponential, trigonometric, and inverse trigonometric functions.

#Function TypeDifferential FormulaIntegral FormulaNotes
1Power Function(d/dx) xⁿ = n xⁿ⁻¹∫ xⁿ dx = xⁿ⁺¹ / (n + 1)n ≠ -1
2Logarithmic(d/dx) ln(x) = 1/x∫ 1/x dx = ln(x)x > 0
3Exponential (base e)(d/dx) eᵃˣ = a eᵃˣ∫ eᵃˣ dx = eᵃˣ / aa ≠ 0
4Exponential (base a)(d/dx) aˣ = aˣ ln(a)∫ aˣ dx = aˣ / ln(a)a > 0, a ≠ 1
5Power Exponential(d/dx) xˣ = xˣ(1 + ln(x))N/ASpecial case
6Logarithm of xN/A∫ ln(x) dx = x(ln(x) - 1)Integration by parts
7Sine(d/dx) sin(x) = cos(x)∫ sin(x) dx = -cos(x)Standard trig
8Cosine(d/dx) cos(x) = -sin(x)∫ cos(x) dx = sin(x)Standard trig
9Tangent(d/dx) tan(x) = sec²(x)∫ tan(x) dx = -ln(cos(x))Trigonometric
10Cotangent(d/dx) cot(x) = -cosec²(x)∫ cot(x) dx = ln(sin(x))Trigonometric
11Secant SquaredN/A∫ sec²(x) dx = tan(x)Trig derivative inverse
12Cosecant SquaredN/A∫ cosec²(x) dx = -cot(x)Trig derivative inverse
13Inverse Sine(d/dx) sin⁻¹(x) = 1 / √(1 - x²)∫ 1/√(1 - x²) dx = sin⁻¹(x)|x| < 1
14Inverse Cosine(d/dx) cos⁻¹(x) = -1 / √(1 - x²)N/A|x| < 1
15Inverse Tangent(d/dx) tan⁻¹(x) = 1 / (1 + x²)∫ 1/(1 + x²) dx = tan⁻¹(x)All x
16Inverse Cotangent(d/dx) cot⁻¹(x) = -1 / (1 + x²)N/AAll x