Calculus^[1]

If f and g are functions of x and a and b are constants, then:   ${\displaystyle {\frac {d}{dx}}x^{n}=nx^{n-1}.}$           

${\displaystyle {\frac {d(af+bg)}{dx}}=a{\frac {df}{dx}}+b{\frac {dg}{dx}}.}$           ${\displaystyle {\frac {d(fg)}{dx}}={\frac {df}{dx}}g+f{\frac {dg}{dx}}.}$           

${\displaystyle {\frac {dh}{dx}}={\frac {dh}{dg}}{\frac {dg}{dx}}.}$           ${\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-g'f}{g^{2}}}.}$

If y=y(x) and x=x(y) are inverse functions then: ${\displaystyle {\frac {dx}{dy}}={\frac {1}{dy/dx}}.}$

Indefinite integrals, where ${\displaystyle C}$ is the arbitrary constant of integration:

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

${\displaystyle \int \!x^{-1}\,dx=\ln |x|+C,}$

Exponential and trigonometric functions

If a is a constant, then: ${\displaystyle {\frac {d}{dx}}\left(e^{ax}\right)=ae^{ax}.}$             ${\displaystyle {\frac {d}{dx}}\left(\ln x\right)={1 \over x},\quad x\neq 0}$    ${\displaystyle \Rightarrow (\ln f)'={\frac {f'}{f}}\quad }$ wherever f is positive.

${\displaystyle (\sin ax)'=a\cos x\,}$	${\displaystyle (\arcsin x)'={1 \over {\sqrt {1-x^{2}}}}\,}$
${\displaystyle (\cos ax)'=-a\sin x\,}$	${\displaystyle (\arccos x)'=-{1 \over {\sqrt {1-x^{2}}}}\,}$
${\displaystyle (\tan x)'=\sec ^{2}x={1 \over \cos ^{2}x}=1+\tan ^{2}x\,}$	${\displaystyle (\arctan x)'={1 \over 1+x^{2}}\,}$
${\displaystyle (\sec x)'=\sec x\tan x\,}$	${\displaystyle (\operatorname {arcsec} x)'={1 \over |x|{\sqrt {x^{2}-1}}}\,}$
${\displaystyle (\csc x)'=-\csc x\cot x\,}$	${\displaystyle (\operatorname {arccsc} x)'=-{1 \over |x|{\sqrt {x^{2}-1}}}\,}$
${\displaystyle (\cot x)'=-\csc ^{2}x={-1 \over \sin ^{2}x}=-(1+\cot ^{2}x)\,}$	${\displaystyle (\operatorname {arccot} x)'=-{1 \over 1+x^{2}}\,}$

Fundamental theorem of calculus

${\displaystyle \int _{a}^{b}{\frac {dF}{ds}}\,ds=\int dF=F|_{a}^{b}=F(b)-F(a)}$

${\displaystyle \Rightarrow {\text{If }}\;F(x)=\int _{a}^{x}f(s)\,ds,\,}$   ${\displaystyle {\text{ then }}\;{\frac {dF}{dx}}=f(x)}$

Taylor series and Euler's equations^[2]

${\displaystyle e^{x}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\dots }$

${\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots }$

${\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots }$

${\displaystyle \Rightarrow \;e^{i\theta }=\cos \theta +i\sin \theta }$