Calculus Sheet (Accessible HTML)

The derivative of c times f of x, equals c times f prime of x, where c is any constant. \[ \dfrac{d}{dx}\!\left(cf(x)\right) = cf'(x) \quad \text{c is any constant} \]

The derivative of f of x plus or minus g of x, equals f prime of x plus or minus g prime of x. \[ \left(f(x) \pm g(x)\right)' = f'(x) \pm g'(x) \]

The derivative of x to the power of n, equals n times x to the power of n minus 1, where n is any number. \[ \dfrac{d}{dx}\!\left(x^n\right) = nx^{n-1} \quad \text{n is any number} \]

The derivative of c equals 0, where c is any constant. \[ \dfrac{d}{dx}\!\left(c\right) = 0 \quad \text{c is any constant} \]

Product Rule: The derivative of f times g equals f prime times g, plus f times g prime. \[ \text{Product Rule:} \quad (fg)' = f'g + fg' \]

Quotient Rule: The derivative of f over g equals f prime times g minus f times g prime, all over g squared. \[ \text{Quotient Rule:} \quad \left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2} \]

Chain Rule: The derivative of f of g of x equals f prime of g of x, times g prime of x. \[ \text{Chain Rule:} \quad \dfrac{d}{dx}\!\left[f(g(x))\right] = f'\!\left(g(x)\right)g'(x) \]

The derivative of e to the power of g of x equals g prime of x times e to the power of g of x. \[ \dfrac{d}{dx}\!\left(e^{g(x)}\right) = g'(x)\,e^{g(x)} \]

The derivative of the natural log of g of x equals g prime of x over g of x. \[ \dfrac{d}{dx}\!\left(\ln g(x)\right) = \dfrac{g'(x)}{g(x)} \]

The derivative of c equals 0.\[ \dfrac{d}{dx}(c) = 0 \]

The derivative of x equals 1.\[ \dfrac{d}{dx}(x) = 1 \]

The derivative of c times x equals c.\[ \dfrac{d}{dx}(cx) = c \]

The derivative of x to the power of n equals n times x to the power of n minus 1.\[ \dfrac{d}{dx}(x^n) = nx^{n-1} \]

The derivative of c times x to the power of n equals n times c times x to the power of n minus 1.\[ \dfrac{d}{dx}(cx^n) = ncx^{n-1} \]

The derivative of sine of x equals cosine of x.\[ \dfrac{d}{dx}(\sin x) = \cos x \]

The derivative of cosine of x equals negative sine of x.\[ \dfrac{d}{dx}(\cos x) = -\sin x \]

The derivative of tangent of x equals secant squared of x.\[ \dfrac{d}{dx}(\tan x) = \sec^2 x \]

The derivative of cotangent of x equals negative cosecant squared of x.\[ \dfrac{d}{dx}(\cot x) = -\csc^2 x \]

The derivative of secant of x equals secant of x times tangent of x.\[ \dfrac{d}{dx}(\sec x) = \sec x\tan x \]

The derivative of cosecant of x equals negative cosecant of x times cotangent of x.\[ \dfrac{d}{dx}(\csc x) = -\csc x\cot x \]

The derivative of inverse sine of x equals 1 over the square root of 1 minus x squared.\[ \dfrac{d}{dx}(\sin^{-1} x) = \dfrac{1}{\sqrt{1-x^2}} \]

The derivative of inverse cosine of x equals negative 1 over the square root of 1 minus x squared.\[ \dfrac{d}{dx}(\cos^{-1} x) = \dfrac{-1}{\sqrt{1-x^2}} \]

The derivative of inverse tangent of x equals 1 over the quantity 1 plus x squared.\[ \dfrac{d}{dx}(\tan^{-1} x) = \dfrac{1}{1+x^2} \]

The derivative of inverse cotangent of x equals negative 1 over the quantity 1 plus x squared.\[ \dfrac{d}{dx}(\cot^{-1} x) = \dfrac{-1}{1+x^2} \]

The derivative of inverse secant of x equals 1 over the absolute value of x times the square root of x squared minus 1.\[ \dfrac{d}{dx}(\sec^{-1} x) = \dfrac{1}{|x|\sqrt{x^2-1}} \]

The derivative of inverse cosecant of x equals negative 1 over the absolute value of x times the square root of x squared minus 1.\[ \dfrac{d}{dx}(\csc^{-1} x) = \dfrac{-1}{|x|\sqrt{x^2-1}} \]

The derivative of a to the power of x equals a to the power of x times the natural log of a.\[ \dfrac{d}{dx}(a^x) = a^x \ln(a) \]

The derivative of e to the power of x equals e to the power of x.\[ \dfrac{d}{dx}(e^x) = e^x \]

The derivative of the natural log of x equals 1 over x, for x greater than 0.\[ \dfrac{d}{dx}(\ln x) = \dfrac{1}{x}, \quad x > 0 \]

The derivative of the natural log of the absolute value of x equals 1 over x, for x not equal to 0.\[ \dfrac{d}{dx}(\ln|x|) = \dfrac{1}{x}, \quad x \neq 0 \]

The derivative of log base a of x equals 1 over the quantity x times the natural log of a, for x greater than 0.\[ \dfrac{d}{dx}(\log_a x) = \dfrac{1}{x \ln a}, \quad x > 0 \]

The derivative of hyperbolic sine of x equals hyperbolic cosine of x.\[ \dfrac{d}{dx}(\sinh x) = \cosh x \]

The derivative of hyperbolic cosine of x equals hyperbolic sine of x.\[ \dfrac{d}{dx}(\cosh x) = \sinh x \]

The derivative of hyperbolic tangent of x equals hyperbolic secant squared of x.\[ \dfrac{d}{dx}(\tanh x) = \operatorname{sech}^2 x \]

The derivative of hyperbolic cotangent of x equals negative hyperbolic cosecant squared of x.\[ \dfrac{d}{dx}(\coth x) = -\operatorname{csch}^2 x \]

The derivative of hyperbolic secant of x equals negative hyperbolic secant of x times hyperbolic tangent of x.\[ \dfrac{d}{dx}(\operatorname{sech} x) = -\operatorname{sech} x \tanh x \]

The derivative of hyperbolic cosecant of x equals negative hyperbolic cosecant of x times hyperbolic cotangent of x.\[ \dfrac{d}{dx}(\operatorname{csch} x) = -\operatorname{csch} x \coth x \]

The integral of c times f of x d x, equals c times the integral of f of x d x, where c is a constant.\[ \int cf(x)\,dx = c\int f(x)\,dx \quad \text{c is a constant} \]

The integral of f of x plus or minus g of x d x, equals the integral of f of x d x, plus or minus the integral of g of x d x.\[ \int \left(f(x) \pm g(x)\right)\,dx = \int f(x)\,dx \pm \int g(x)\,dx \]

The definite integral from a to b of f of x d x, equals F of x evaluated from a to b, which equals F of b minus F of a, where F of x is the integral of f of x d x.\[ \int_a^b f(x)\,dx = F(x)\Big|_a^b = F(b)-F(a) \quad \text{where } F(x) = \int f(x)\,dx \]

The definite integral from a to b of c times f of x d x, equals c times the definite integral from a to b of f of x d x.\[ \int_a^b cf(x)\,dx = c\int_a^b f(x)\,dx \]

The definite integral from a to a of f of x d x equals 0. The definite integral from a to b of f of x d x equals negative the definite integral from b to a of f of x d x.\[ \int_a^a f(x)\,dx = 0 \qquad \int_a^b f(x)\,dx = -\int_b^a f(x)\,dx \]

The definite integral from a to b of f of x d x, equals the definite integral from a to c of f of x d x, plus the definite integral from c to b of f of x d x.\[ \int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx \]

The definite integral from a to b of c d x, equals c times the quantity b minus a.\[ \int_a^b c\,dx = c(b-a) \]

If f of x is greater than or equal to 0 on the interval a to b, then the integral from a to b of f of x d x is greater than or equal to 0.\[ \text{If } f(x) \geq 0 \text{ on } a \leq x \leq b, \text{ then } \int_a^b f(x)\,dx \geq 0 \]

If f of x is greater than or equal to g of x on the interval a to b, then the integral from a to b of f of x d x is greater than or equal to the integral from a to b of g of x d x.\[ \text{If } f(x) \geq g(x) \text{ on } a \leq x \leq b, \text{ then } \int_a^b f(x)\,dx \geq \int_a^b g(x)\,dx \]

The integral of 1 d x equals x plus c.\[ \int dx = x + c \]

The integral of k d x equals k x plus c.\[ \int k\,dx = kx + c \]

The integral of x to the power of n d x, equals 1 over the quantity n plus 1, times x to the power of n plus 1, plus c, for n not equal to negative 1.\[ \int x^n\,dx = \dfrac{1}{n+1}x^{n+1}+c, \quad n \neq -1 \]

The integral of 1 over x d x, equals the natural log of the absolute value of x, plus c.\[ \int \dfrac{1}{x}\,dx = \ln|x|+c \]

The integral of 1 over the quantity a x plus b d x, equals 1 over a times the natural log of the absolute value of a x plus b, plus c.\[ \int \dfrac{1}{ax+b}\,dx = \dfrac{1}{a}\ln|ax+b|+c \]

The integral of x to the power of p over q d x, equals q over the quantity p plus q, times x to the power of the quantity p plus q over q, plus c.\[ \int x^{p/q}\,dx = \dfrac{q}{p+q}\,x^{(p+q)/q}+c \]

The integral of cosine u d u equals sine u plus c.\[ \int \cos u\,du = \sin u + c \]

The integral of sine u d u equals negative cosine u plus c.\[ \int \sin u\,du = -\cos u + c \]

The integral of secant squared u d u equals tangent u plus c.\[ \int \sec^2 u\,du = \tan u + c \]

The integral of cosecant squared u d u equals negative cotangent u plus c.\[ \int \csc^2 u\,du = -\cot u + c \]

The integral of secant u times tangent u d u equals secant u plus c.\[ \int \sec u \tan u\,du = \sec u + c \]

The integral of cosecant u times cotangent u d u equals negative cosecant u plus c.\[ \int \csc u \cot u\,du = -\csc u + c \]

The integral of tangent u d u equals the natural log of the absolute value of secant u plus c.\[ \int \tan u\,du = \ln|\sec u| + c \]

The integral of cotangent u d u equals the natural log of the absolute value of sine u plus c.\[ \int \cot u\,du = \ln|\sin u| + c \]

The integral of secant u d u equals the natural log of the absolute value of secant u plus tangent u plus c.\[ \int \sec u\,du = \ln|\sec u + \tan u| + c \]

The integral of cosecant u d u equals the natural log of the absolute value of cosecant u minus cotangent u plus c.\[ \int \csc u\,du = \ln|\csc u - \cot u| + c \]

The integral of e to the power of u d u equals e to the power of u plus c.\[ \int e^u\,du = e^u + c \]

The integral of a to the power of u d u equals a to the power of u over the natural log of a, plus c.\[ \int a^u\,du = \dfrac{a^u}{\ln a}+c \]

The integral of the natural log of u d u equals u times the natural log of u, minus u, plus c.\[ \int \ln u\,du = u\ln(u) - u + c \]

The integral of u times e to the power of u d u equals the quantity u minus 1 times e to the power of u, plus c.\[ \int u e^u\,du = (u-1)e^u + c \]

The integral of 1 over the quantity u times the natural log of u d u equals the natural log of the absolute value of the natural log of u, plus c.\[ \int \dfrac{1}{u \ln u}\,du = \ln|\ln u|+c \]

The integral of 1 over the square root of a squared minus u squared d u, equals inverse sine of u over a, plus c.\[ \int \dfrac{1}{\sqrt{a^2-u^2}}\,du = \sin^{-1}\!\left(\dfrac{u}{a}\right)+c \]

The integral of 1 over the quantity a squared plus u squared d u, equals 1 over a times inverse tangent of u over a, plus c.\[ \int \dfrac{1}{a^2+u^2}\,du = \dfrac{1}{a}\tan^{-1}\!\left(\dfrac{u}{a}\right)+c \]

The integral of 1 over u times the square root of u squared minus a squared d u, equals 1 over a times inverse secant of u over a, plus c.\[ \int \dfrac{1}{u\sqrt{u^2-a^2}}\,du = \dfrac{1}{a}\sec^{-1}\!\left(\dfrac{u}{a}\right)+c \]

The integral of inverse sine of u d u equals u times inverse sine of u, plus the square root of 1 minus u squared, plus c.\[ \int \sin^{-1}u\,du = u\sin^{-1}u + \sqrt{1-u^2}+c \]

The integral of inverse tangent of u d u equals u times inverse tangent of u, minus 1 half times the natural log of 1 plus u squared, plus c.\[ \int \tan^{-1}u\,du = u\tan^{-1}u - \tfrac{1}{2}\ln(1+u^2)+c \]

The integral of hyperbolic sine u d u equals hyperbolic cosine u plus c.\[ \int \sinh u\,du = \cosh u+c \]

The integral of hyperbolic cosine u d u equals hyperbolic sine u plus c.\[ \int \cosh u\,du = \sinh u+c \]

The integral of hyperbolic secant squared u d u equals hyperbolic tangent u plus c.\[ \int \operatorname{sech}^2 u\,du = \tanh u+c \]

The integral of hyperbolic cosecant squared u d u equals negative hyperbolic cotangent u plus c.\[ \int \operatorname{csch}^2 u\,du = -\coth u+c \]

The integral of hyperbolic tangent u d u equals the natural log of hyperbolic cosine u plus c.\[ \int \tanh u\,du = \ln(\cosh u)+c \]

Given the integral from a to b of f of g of x times g prime of x d x\(\int_a^b f(g(x))\,g'(x)\,dx\), the substitution u equals g of x\(u = g(x)\) converts it to:

the integral from g of a to g of b of f of u d u\[ \int_{g(a)}^{g(b)} f(u)\,du \]

The integral of u d v equals u times v minus the integral of v d u\[ \int u\,dv = uv - \int v\,du \]

The definite integral from a to b of u d v equals u times v evaluated from a to b, minus the definite integral from a to b of v d u\[ \int_a^b u\,dv = uv\Big|_a^b - \int_a^b v\,du \]

Choose u and dv; compute du by differentiating u; compute v by integrating dv.

If the integral contains the following root, use the given substitution:

The square root of a squared minus b squared x squared, implies x equals a over b times sine theta, where cosine squared theta equals 1 minus sine squared theta\[ \sqrt{a^2 - b^2x^2} \Rightarrow x = \tfrac{a}{b}\sin\theta, \quad \cos^2\theta = 1-\sin^2\theta \]

The square root of b squared x squared minus a squared, implies x equals a over b times secant theta, where tangent squared theta equals secant squared theta minus 1\[ \sqrt{b^2x^2 - a^2} \Rightarrow x = \tfrac{a}{b}\sec\theta, \quad \tan^2\theta = \sec^2\theta-1 \]

The square root of a squared plus b squared x squared, implies x equals a over b times tangent theta, where secant squared theta equals 1 plus tangent squared theta\[ \sqrt{a^2 + b^2x^2} \Rightarrow x = \tfrac{a}{b}\tan\theta, \quad \sec^2\theta = 1+\tan^2\theta \]