# Physics equations/01-Introduction/A:reviewCALCULUS

### 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]

${\displaystyle e^{x}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\dots }$
${\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots }$
${\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots }$
${\displaystyle \Rightarrow \;e^{i\theta }=\cos \theta +i\sin \theta }$
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