Calculus Derivative Rules

A collection of basic derivative rules including power rule, trigonometric functions, inverse trigonometric functions, and algebraic combination rules.

Derivative of $x^n$

$$n x^{n-1}$$

Derivative of $k f(x)$
(Constant Multiplication Rule)

$$k f'(x)$$

Derivative of $f(x) + g(x)$
(Derivative of a sum)

$$f'(x) + g'(x)$$

Derivative of $e^x$

$$e^x$$

Derivative of $a^x$

$$a^{x}{ln}(a)$$

Derivative of ${sin}(x)$

$${cos}(x)$$

Derivative of ${cos}(x)$

$$-{sin}(x)$$

Derivative of ${tan}(x)$

$${sec}^2(x)$$

Derivative of ${cot}(x)$

$$-{csc}^2(x)$$

Derivative of $f(x) g(x)$
(Product Rule)

$$f'g + fg'$$

Derivative of $\frac{f(x)}{g(x)}$
(Quotient Rule)

$$\frac{gf^{\prime}-fg^{\prime}}{g^2}$$

Derivative of $f(g(x))$

(Chain Rule)

$$f'(g(x)) \cdot g'(x)$$

Derivative of $(f^{-1})(x)$

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

Derivative of ${sin}^{-1}(x)$

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

Derivative of ${cos}^{-1}(x)$

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

Derivative of ${tan}^{-1}(x)$

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

Derivative of ${cot}^{-1}(x)$

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

Derivative of ${sec}(x)$

$${sec}(x)t{an}(x)$$

Derivative of ${csc}(x)$

$$-{csc}(x){cot}(x)$$