taylor expansion: $f(x) approx f(x_0) + frac{f’(x_0)}{1!}(x-x_0) + frac{f’‘(x_0)}{2!}(x-x_0)^2 + …$

Single-variable calculus

Derivatives:

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

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

$frac{d}{dx}ln(x) = 1/x$

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

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

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

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

$int tan = ln|sec|$

$int cot = ln|sin|$

$int sec = ln|sec+tan|$

$int csc = ln|csc-cot|$

$int frac{du}{sqrt{a^2-u^2}} = sin^{-1}(frac{u}{a})$

$int frac{du}{usqrt{u^2-a^2}} = frac{1}{a}sec^{-1}(frac{u}{a})$

$int frac{du}{a^2+u^2} = frac{1}{a} tan^{-1}(frac{u}{a})$

Continuous: left limit = right limit = value

Differentiable: continuous and no sharp points / asymptotes

L’Hospital’s - for indeterminate forms: $(frac{f(x)}{g(x)})’ = frac{f’(x)}{g’(x)}$

Integration by parts: $int{udv}=uv-int{duv}$, LIATE

Expansions:

$e^x = sum{frac{x^n}{n!}}$

$sin(x) = sum_0^infty{frac{(-1)^n x^{2n+1}}{(2n+1)!}}$

$cos(x) = sum_0^infty{frac{(-1)^n x^{2n}}{(2n)!}}$

Geometric Sum: $a_{1st}frac{1-r^{n+1}}{1-r}$

Multivariable calculus