Calculus

Basic Differentiation Rules

$\frac{d}{dx}\left[cu\right]=cu´$
$\frac{d}{dx}\left[u±v\right]=u´±v´$
$\frac{d}{dx}\left[uv\right]=uv´+vu´$
$\frac{d}{dx}\left[\frac{u}{v}\right]=\frac{vu´+uv´}{{v}^{2}}$
$\frac{d}{dx}\left[c\right]=0$
$\frac{d}{dx}\left[{u}^{n}\right]=n{u}^{n-1}u´$
$\frac{d}{dx}\left[x\right]=1$
$\frac{d}{dx}\left[\mathrm{ln}u\right]=\frac{u´}{u}$
$\frac{d}{dx}\left[{e}^{u}\right]={e}^{u}u´$
$\frac{d}{dx}\left[{\mathrm{log}}_{a}u\right]=\frac{u´}{\left(\mathrm{ln}a\right)u}$
$\frac{d}{dx}\left[{a}^{u}\right]=\left(\mathrm{ln}a\right){a}^{u}u´$
$\frac{d}{dx}\left[\mathrm{sin}u\right]=\left(\mathrm{cos}u\right)u´$
$\frac{d}{dx}\left[\mathrm{cos}u\right]=-\left(\mathrm{sin}u\right)u´$

Basic Integration Rules (a > 0)

$\underset{\phantom{﻿}}{\overset{\phantom{﻿}}{\int }}du=u+C$
$\underset{\phantom{﻿}}{\overset{\phantom{﻿}}{\int }}\frac{du}{u}=\mathrm{\mathrm{l}\mathrm{n}}|u|+C$
$\underset{\phantom{﻿}}{\overset{\phantom{﻿}}{\int }}{e}^{u}du={e}^{u}+C$
$\underset{\phantom{﻿}}{\overset{\phantom{﻿}}{\int }}{a}^{u}du=\left(\frac{1}{\mathrm{\mathrm{l}\mathrm{n}}a}\right){a}^{u}+C$
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