Calculus Formulas

$\lim_{x\to a} [f(x)+g(x)] = \lim_{x\to a}f(x)+\lim_{x\to a}g(x)$

$\lim_{x\to a} [f(x)\cdot g(x)] = \lim_{x\to a}f(x)\cdot\lim_{x\to a}g(x)$

$\lim_{x\to 0}\frac{\sin x}{x}=1$

a standard limit used to derive $\frac{d}{dx}[\sin x]$

$\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x=e$

one definition of Euler's number $e$

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

power rule

$\frac{d}{dx}[fg]=f'g+fg'$

product rule

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

quotient rule

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

chain rule

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

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

$x > 0$

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

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

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

$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$

power rule — $n \neq -1$

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

$\int e^x\,dx=e^x+C$

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

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

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

integration by parts

$\int_a^b f(x)\,dx=F(b)-F(a)$

where $F'(x)=f(x)$

$\frac{d}{dx}\int_a^x f(t)\,dt=f(x)$

$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$

Taylor series, centered at $x=a$

$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$

Maclaurin series — Taylor series centered at $x=0$