Derivative Rules

A comprehensive collection of vocabulary-style flashcards covering standard calculus derivative rules, including basic algebraic rules, exponential, logarithmic, trigonometric, and inverse trigonometric derivatives.

Derivative of constant

$$\frac{d}{dx}(c) = 0$$

Derivative of power rule

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

Derivative of product rule

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

Derivative of quotient rule

$$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x) \times g(x) - f(x) \times g'(x)}{(g(x))^2}$$

Derivative of chain rule

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

Derivative of chain rule (reciprocal of a function)

$$\frac{d}{dx}\left(\frac{1}{g(x)}\right) = -\frac{g'(x)}{(g(x))^2}$$

Derivative of inverse

$$\frac{d}{dx}(f^{-1}(x)) = \frac{1}{f'(f^{-1}(x))}$$

Derivative of exponential function ($e^x$)

$$\frac{d}{dx}(e^x) = e^x$$

Derivative of exponential function ($b^x$)

$\frac{d}{dx}(b^x) = b^x \ln(b)$, where $b = \text{constant}$

Derivative of sine function

$$\frac{d}{dx}(\sin(x)) = \cos(x)$$

Derivative of cosine function

$$\frac{d}{dx}(\cos(x)) = -\sin(x)$$

Derivative of tan function

$$\frac{d}{dx}(\tan(x)) = \sec^2(x)$$

Derivative of secant function

$$\frac{d}{dx}(\sec(x)) = \sec(x)\tan(x)$$

Derivative of cosecant function

$$\frac{d}{dx}(\csc(x)) = -\csc(x)\cot(x)$$

Derivative of cotangent function

$$\frac{d}{dx}(\cot(x)) = -\csc^2(x)$$

Derivative of natural log

$$\frac{d}{dx}(\ln(x)) = \frac{1}{x}$$

Derivative of logarithm

$$\frac{d}{dx}(\log_a(x)) = \frac{1}{x\ln(a)}$$

Derivative of inverse sine function

$$\frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1-x^2}}$$

Derivative of inverse cosine function

$$\frac{d}{dx}(\cos^{-1}(x)) = -\frac{1}{\sqrt{1-x^2}}$$

Derivative of inverse tan function

$$\frac{d}{dx}(\tan^{-1}(x)) = \frac{1}{1+x^2}$$

Derivative of inverse secant function

$$\frac{d}{dx}(\sec^{-1}(x)) = \frac{1}{x\sqrt{x^2-1}}$$

Derivative of inverse cosecant function

$$\frac{d}{dx}(\csc^{-1}(x)) = -\frac{1}{x\sqrt{x^2-1}}$$

Derivative of inverse cotangent function

$$\frac{d}{dx}(\cot^{-1}(x)) = -\frac{1}{1+x^2}$$