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Section B.1 Differentiation Formulas
List B.1.1. Derivative Rules
\(\displaystyle \lzo{x}(cx)=c\)
\(\displaystyle \lzo{x}(u\pm v)=u'\pm v'\)
\(\displaystyle \lzo{x}(u\cdot v)=uv'+ u'v\)
\(\displaystyle \lzo{x}(\frac uv)=\frac{vu'-uv'}{v^2}\)
\(\displaystyle \lzo{x}(u(v))=u'(v)v'\)
\(\displaystyle \lzo{x}(c)=0\)
\(\displaystyle \lzo{x}(x)=1\)
\(\displaystyle \lzo{x}(x^n)=nx^{n-1}\)
\(\displaystyle \lzo{x}(e^x)=e^x\)
\(\displaystyle \lzo{x}(a^x)=\ln a\cdot a^x\)
\(\displaystyle \lzo{x}(\ln x)=\frac{1}{x}\)
\(\displaystyle \lzo{x}(\log_a x)=\frac{1}{\ln a}\cdot \frac1x\)
\(\displaystyle \lzo{x}(\sin(x))=\cos(x)\)
\(\displaystyle \lzo{x}(\cos(x))=-\sin(x)\)
\(\displaystyle \lzo{x}(\csc(x))=-\csc(x)\cot(x)\)
\(\displaystyle \lzo{x}(\sec(x))=\sec(x)\tan(x)\)
\(\displaystyle \lzo{x}(\tan(x))=\sec^2(x)\)
\(\displaystyle \lzo{x}(\cot(x))=-\csc^2(x)\)
\(\displaystyle \lzo{x}(\cosh=(x))=\sinh(x)\)
\(\displaystyle \lzo{x}(\sinh(x))=\cosh(x)\)
\(\displaystyle \lzo{x}(\sech(x))=-\sech(x)\tanh(x)\)
\(\displaystyle \lzo{x}(\tanh(x))=\sech^2(x)\)
\(\displaystyle \lzo{x}(\csch(x))=-\csch(x)\coth(x)\)
\(\displaystyle \lzo{x}(\coth(x))=-\csch^2(x)\)
\(\displaystyle \lzo{x}(\sin^{-1}(x))=\frac{1}{\sqrt{1-x^2}}\)
\(\displaystyle \lzo{x}(\cos^{-1}(x))=\frac{-1}{\sqrt{1-x^2}}\)
\(\displaystyle \lzo{x}(\csc^{-1}(x))=\frac{-1}{\abs{x}\sqrt{x^2-1}}\)
\(\displaystyle \lzo{x}(\sec^{-1}(x))=\frac{1}{\abs{x}\sqrt{x^2-1}}\)
\(\displaystyle \lzo{x}(\tan^{-1}(x))=\frac{1}{1+x^2}\)
\(\displaystyle \lzo{x}(\cot^{-1}(x))=\frac{-1}{1+x^2}\)
\(\displaystyle \lzo{x}(\cosh^{-1}(x))=\frac1{\sqrt{x^2-1}}\)
\(\displaystyle \lzo{x}(\sinh^{-1}(x))=\frac1{\sqrt{x^2+1}}\)
\(\displaystyle \lzo{x}(\sech^{-1}(x))=\frac{-1}{x\sqrt{1-x^2}}\)
\(\displaystyle \lzo{x}(\csch^{-1}(x))=\frac{-1}{\abs{x}\sqrt{1+x^2}}\)
\(\displaystyle \lzo{x}(\tanh^{-1}(x))=\frac1{1-x^2}\)
\(\displaystyle \lzo{x}(\coth^{-1}(x))=\frac1{1-x^2}\)
