Calculus: Differentiation and Integration Formulas

A collection of core differentiation and integration formulas including trigonometric, exponential, logarithmic, and inverse trigonometric functions.

What is the derivative of a constant $k$?

$$\frac{d}{dx}k = 0$$

What is the product rule for differentiation?

$$\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)$$

What is the quotient rule for differentiation?

$$\frac{d}{dx}\frac{f(x)}{g(x)} = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$$

What is the chain rule for the composition of functions $f(g(x))$?

$$\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$$

What is the power rule for differentiation?

$$\frac{d}{dx}x^n = nx^{n-1}$$

What is the derivative of $\sin(x)$?

$$\cos(x)$$

What is the derivative of $\tan(x)$?

$$\sec^2(x)$$

What is the derivative of $\sec(x)$?

$$\sec(x)\tan(x)$$

What is the derivative of $a^x$?

$$a^x \ln(a)$$

What is the derivative of $\ln|x|$?

$$\frac{1}{x}$$

What is the derivative of $\sin^{-1}(x)$?

$$\frac{1}{\sqrt{1-x^2}}$$

What is the derivative of $\tan^{-1}(x)$?

$$\frac{1}{x^2+1}$$

What is the integral of $x^n$ with respect to $x$?

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$

What is the integral of $\frac{1}{x}$?

$$\ln|x| + C$$

What is the integral of $a^x$?

$$\frac{1}{\ln(a)} a^x + C$$

What is the integral of $\ln(x)$?

$$x \ln(x) - x + C$$

What is the integral of $\sin(x)$?

$$-\cos(x) + C$$

What is the integral of $\tan(x)$?

$$-\ln|\cos(x)| + C$$

What is the integral of $\sec(x)$?

$$\ln|\sec(x) + \tan(x)| + C$$

What is the integral of $\sec^2(x)$?

$$\tan(x) + C$$

What is the integral of $\frac{1}{a^2 + x^2}$?

$$\frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$$

What is the integral of $\frac{1}{\sqrt{a^2 - x^2}}$?

$$\sin^{-1}(\frac{x}{a}) + C$$