Physics equations/01-Introduction/A:reviewCALCULUS

Calculus

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

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

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

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

Indefinite integrals, where $C$ is the arbitrary constant of integration:

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

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

Exponential and trigonometric functions

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

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

Fundamental theorem of calculus

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

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

Taylor series and Euler's equations

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

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

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

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