# Derivatives / Differential Calculus

Formulae on Differentiation.

#### Derivatives

$$\frac{d}{dx}C = 0, where\quad C = constat$$

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

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

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

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

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

$$\frac{d}{dx}a^x = a^x\ln{a} $$

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

$$\frac{d}{dx}\sinh{x} = \cosh{x} $$

$$\frac{d}{dx}\cosh{x} = \sinh{x} $$

$$\frac{d}{dx}\sinh^{-1}{x} = \frac{1}{\sqrt{x^2 + 1}}$$

$$\frac{d}{dx}\cosh^{-1}{x} = \frac{1}{\sqrt{x^2 - 1}} $$

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

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

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

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

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

$$\frac{d}{dx}\tanh{x} = \frac{1}{\cosh^{2}{x}} $$

$$\frac{d}{dx}\coth{x} = \frac{1}{\sinh^{2}{x}} $$

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

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

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

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