APEX Differentiation Formulas

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Section B.1 Differentiation Formulas

List B.1.1. Derivative Rules

\(\displaystyle \lzo{x}(cx)=c\)
\(\displaystyle \lzo{x}(u\pm v)=u'\pm v'\)
\(\displaystyle \lzo{x}(u\cdot v)=uv'+ u'v\)
\(\displaystyle \lzo{x}(\frac uv)=\frac{vu'-uv'}{v^2}\)
\(\displaystyle \lzo{x}(u(v))=u'(v)v'\)
\(\displaystyle \lzo{x}(c)=0\)
\(\displaystyle \lzo{x}(x)=1\)

List B.1.2. Derivatives of Elementary Functions

\(\displaystyle \lzo{x}(x^n)=nx^{n-1}\)
\(\displaystyle \lzo{x}(e^x)=e^x\)
\(\displaystyle \lzo{x}(a^x)=\ln a\cdot a^x\)
\(\displaystyle \lzo{x}(\ln x)=\frac{1}{x}\)
\(\displaystyle \lzo{x}(\log_a x)=\frac{1}{\ln a}\cdot \frac1x\)
\(\displaystyle \lzo{x}(\sin x)=\cos x\)
\(\displaystyle \lzo{x}(\cos x)=-\sin x\)
\(\displaystyle \lzo{x}(\csc x)=-\csc x\cot x\)
\(\displaystyle \lzo{x}(\sec x)=\sec x\tan x\)
\(\displaystyle \lzo{x}(\tan x)=\sec^2 x\)
\(\displaystyle \lzo{x}(\cot x)=-\csc^2 x\)
\(\displaystyle \lzo{x}(\cosh x)=\sinh x\)
\(\displaystyle \lzo{x}(\sinh x)=\cosh x\)
\(\displaystyle \lzo{x}(\tanh x)=\sech^2 x\)
\(\displaystyle \lzo{x}(\sech x)=-\sech x\tanh x\)
\(\displaystyle \lzo{x}(\coth x)=-\csch^2 x\)
\(\displaystyle \lzo{x}(\csch x)=-\csch x\coth x\)

List B.1.3. Derivatives of Inverse Functions

\(\displaystyle \lzo{x}(\sin^{-1}x)=\frac{1}{\sqrt{1-x^2}}\)
\(\displaystyle \lzo{x}(\cos^{-1}x)=\frac{-1}{\sqrt{1-x^2}}\)
\(\displaystyle \lzo{x}(\csc^{-1}x)=\frac{-1}{\abs{x}\sqrt{x^2-1}}\)
\(\displaystyle \lzo{x}(\sec^{-1}x)=\frac{1}{\abs{x}\sqrt{x^2-1}}\)
\(\displaystyle \lzo{x}(\tan^{-1}x)=\frac{1}{1+x^2}\)
\(\displaystyle \lzo{x}(\cot^{-1}x)=\frac{-1}{1+x^2}\)
\(\displaystyle \lzo{x}(\cosh^{-1}x)=\frac1{\sqrt{x^2-1}}\)
\(\displaystyle \lzo{x}(\sinh^{-1}x)=\frac1{\sqrt{x^2+1}}\)
\(\displaystyle \lzo{x}(\sech^{-1}x)=\frac{-1}{x\sqrt{1-x^2}}\)
\(\displaystyle \lzo{x}(\csch^{-1}x)=\frac{-1}{\abs{x}\sqrt{1+x^2}}\)
\(\displaystyle \lzo{x}(\tanh^{-1}x)=\frac1{1-x^2}\)
\(\displaystyle \lzo{x}(\coth^{-1}x)=\frac1{1-x^2}\)