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Subsection A.5 Differentiation Rules
Subsubsection A.5.1 Basic Rules
\(\displaystyle \displaystyle \frac{d}{dx}\left(cx\right)=c\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(u\pm v\right)=u^{\prime }\pm v^{\prime }\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(u\cdot v\right)=uv^{\prime }+ u^{\prime }v\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\frac{u}{v}\right)=\frac{vu^{\prime }-uv^{\prime }}{v^2}\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(u(v)\right)=u^{\prime }(v)v^{\prime }\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(c\right)=0\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(x\right)=1\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(x^n\right)=nx^{n-1}\)
Subsubsection A.5.2 Exponential and Logarithmic Functions
\(\displaystyle \displaystyle \frac{d}{dx}\left(e^x\right)=e^x\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(a^x\right)=\ln a\cdot a^x\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\ln x\right)=\frac{1}{x}\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\log _a x\right)=\frac{1}{\ln a}\cdot \frac{1}{x}\)
Subsubsection A.5.3 Trigonometric Functions
\(\displaystyle \displaystyle \frac{d}{dx}\left(\sin x\right)=\cos x\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\cos x\right)=-\sin x\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\csc x\right)=-\csc x\cot x\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\sec x\right)=\sec x\tan x\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\tan x\right)=\sec ^2 x\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\cot x\right)=-\csc ^2 x\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\sin ^{-1}x\right)=\frac{1}{\sqrt{1-x^2}}\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\cos ^{-1}x\right)=\frac{-1}{\sqrt{1-x^2}}\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\csc ^{-1}x\right)=\frac{-1}{x\sqrt{x^2-1}}\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\sec ^{-1}x\right)=\frac{1}{x\sqrt{x^2-1}}\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\tan ^{-1}x\right)=\frac{1}{1+x^2}\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\cot ^{-1}x\right)=\frac{-1}{1+x^2}\)
Subsubsection A.5.4 Hyperbolic Trigonometric Functions
\(\displaystyle \displaystyle \frac{d}{dx}\left(\cosh x\right)=\sinh x\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\sinh x\right)=\cosh x\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\tanh x\right)=\operatorname{sech}^2 x\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\operatorname{sech}x\right)=-\operatorname{sech}x\tanh x\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\operatorname{csch}x\right)=-\operatorname{csch}x\coth x\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\coth x\right)=-\operatorname{csch}^2 x\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\cosh ^{-1}x\right)=\frac{1}{\sqrt{x^2-1}}\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\sinh ^{-1}x\right)=\frac{1}{\sqrt{x^2+1}}\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\operatorname{sech}^{-1}x\right)=\frac{-1}{x\sqrt{1-x^2}}\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\operatorname{csch}^{-1}x\right)=\frac{-1}{|x|\sqrt{1+x^2}}\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\tanh ^{-1}x\right)=\frac{1}{1-x^2}\)
\(\displaystyle \displaystyle \frac{d}{dx}\left(\coth ^{-1}x\right)=\frac{1}{1-x^2}\)
