Physics equations/01-Introduction/A:reviewCALCULUS

Calculus

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

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

$\frac{dh}{dx}=\frac{dh}{dg}\frac{dg}{dx}.$ Template:Spaces$\left(\frac{f}{g}\right)=\frac{f^{\prime }g-g^{\prime }f}{g^2}.$

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:

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

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

Exponential and trigonometric functions

If a is a constant, then: $\frac{d}{dx}\left(e^{ax}\right)=a{e}^{ax}.$ Template:Spaces$\frac{d}{dx}(\ln x)=\frac{1}{x},x\ne 0$ Template:Spaces$\Rightarrow (\ln f{)}^{\prime }=\frac{f^{\prime }}{f}$ wherever f is positive.

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

Fundamental theorem of calculus

$\underset{a}{\overset{b}{\int }}\frac{dF}{ds}ds=\int dF=F{|}_a^b=F(b)-F(a)$

$\Rightarrow \text{If }F(x)=\underset{a}{\overset{x}{\int }}f(s)ds,$Template:Spaces $\text{ then }\frac{dF}{dx}=f(x)$

Taylor series and Euler's equations

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

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

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

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

• Derivative
• Differential calculus
• Vector calculus identities
• Differentiable function
• Differential of a function
• Limit of a function
• Function
• List of mathematical functions
• Trigonometric functions
• Inverse trigonometric functions
• Hyperbolic functions
• Inverse hyperbolic functions
• Matrix calculus
• Differentiation under the integral sign