deriv and integration rules

Power Rule for Derivatives

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

Product Rule for Derivatives

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

Quotient Rule for Derivatives

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

Chain Rule for Derivatives

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

Derivative of $a^x$

$$\frac{d}{dx}[a^x] = a^x \ln(a)$$

Derivative of $\log_a(x)$

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

Derivative of $\sin(x)$

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

Derivative of $\cos(x)$

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

Derivative of $\tan(x)$

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

Derivative of $\csc(x)$

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

Derivative of $\sec(x)$

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

Derivative of $\cot(x)$

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

Derivative of $\arcsin(x)$

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

Derivative of $\arccos(x)$

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

Derivative of $\arctan(x)$

$$\frac{d}{dx}[\arctan(x)] = \frac{1}{1 + x^2}$$

Power Rule for Integration

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ (where $n \neq -1$)

Integral of $\frac{1}{x}$

$$\int \frac{1}{x}\,dx = \ln(|x|) + C$$

Integral of $a^x$

$$\int a^x\,dx = \frac{a^x}{\ln(a)} + C$$

Integration by Parts Formula

$$\int u\,dv = u v - \int v\,du$$

Integral of $\sin(x)$

$$\int \sin(x)\,dx = -\cos(x) + C$$

Integral of $\cos(x)$

$$\int \cos(x)\,dx = \sin(x) + C$$

Integral of $\tan(x)$

$$\int \tan(x)\,dx = -\ln(|\cos(x)|) + C$$

Integral of $\sec(x)$

$$\int \sec(x)\,dx = \ln(|\sec(x) + \tan(x)|) + C$$

Integral of $\sec^2(x)$

$$\int \sec^2(x)\,dx = \tan(x) + C$$

Integral of $\csc^2(x)$

$$\int \csc^2(x)\,dx = -\cot(x) + C$$

Integral of $\sec(x) \tan(x)$

$$\int \sec(x) \tan(x)\,dx = \sec(x) + C$$

Integral of $\csc(x) \cot(x)$

$$\int \csc(x) \cot(x)\,dx = -\csc(x) + C$$

Integral of $\frac{1}{\sqrt{a^2 - x^2}}$

$$\int \frac{1}{\sqrt{a^2 - x^2}}\,dx = \arcsin\left(\frac{x}{a}\right) + C$$

Integral of $\frac{1}{a^2 + x^2}$

$$\int \frac{1}{a^2 + x^2}\,dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$$