Derivative Rules (proper formatting!)
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Description and Tags
Various derivative rules/laws with proper formatting
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21 Terms
\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
