Derivative Rules (proper formatting!)

Various derivative rules/laws with proper formatting

\frac{d}{dx}\ln x

\frac{1}{x}

\frac{d}{dx}\left(\sin^{-1}x\right)=

\frac{1}{\sqrt{1-x^2}},x\ne\pm1

\frac{d}{dx}\left(\cos^{-1}x\right)=

\frac{-1}{\sqrt{1-x^2}},x\ne\pm1

\frac{d}{dx}\left(\tan^{-1}x\right)=

\frac{1}{1+x^2}

\frac{d}{dx}\left(\cot^{-1}x\right)=

\frac{-1}{1+x^2}

\frac{d}{dx}\left(\sec^{-1}x\right)=

\frac{1}{\left|x\right|\sqrt{x^2}-1},x\ne\pm1,0

\frac{d}{dx}\left(\csc^{-1}x\right)=

\frac{-1}{\left|x\right|\sqrt{x^2}-1},x\ne\pm1,0

\frac{d}{dx}\left(a^{x}\right)=

\ln\left(a\right)a^{x}

\frac{d}{dx}\log_{a}x=

\frac{1}{x\ln a}

Power Rule

\frac{d}{dx}\left(x^{n}\right)=nx^{n-1}

Product Rule

\frac{d}{dx}\left(uv\right)=u^{\prime}v+uv^{\prime}

Quotient Rule

\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u^{\prime}v-uv^{\prime}}{v^2}

Chain Rule

\frac{d}{dx}\left(f\left(g\left(x\right)\right)\right)=f^{\prime}\left(g\left(x\right)\right)\cdot g^{\prime}\left(x\right)

Definition of a Derivative

f^{\prime}\left(a\right)=\lim_{h\to0}\frac{f\left(a+h\right)-f\left(a\right)}{h}

Alternate Definition of a Derivative

f^{\prime}\left(a\right)=\lim_{x\to a}\frac{f\left(x\right)-f\left(a\right)}{x-a}

\frac{d}{dx}\left(\sin x\right)=

\cos x

\frac{d}{dx}\left(\cos x_{}\right)=

-\sin x

\frac{d}{dx}\left(\tan x\right)=

\sec^2x

\frac{d}{dx}\left(\sec x\right)=

\sec x\tan x

\frac{d}{dx}\left(\csc x\right)=

-\csc x\cdot\cos x

\frac{d}{dx}\left(\cot x\right)=

-\csc^2x