Table of Derivatives

Derivatives of functions

Table of Derivatives

An extened set of derivatives of functions:

\(\frac{d}{dx}a=0\)

\(\frac{d}{dx}x^a=ax^{a-1}\)

\(\frac{d}{dx}e^{ax}=ae^{ax}\)

\(\frac{d}{dx}\ln(ax)=\frac{1}{x}\)

\(\frac{d}{dx}\sin(ax)=a\cos(ax)\)

\(\frac{d}{dx}\cos(ax)=-a\sin(ax)\)

\(\frac{d}{dx}\tan(ax)=a\tan^2(ax)+a=a\sec^2(ax)\)

\(\frac{d}{dx}\cot(ax)=-a\csc^2(ax)\)

\(\frac{d}{dx}\csc(ax)=\frac{-a\cos(ax)}{\sin^2(ax)}=-a\csc(ax)\cot(ax)\)

\(\frac{d}{dx}\arcsin(ax)=\frac{a}{\sqrt{1-a^2x^2}}\)

\(\frac{d}{dx}\arccos(ax)=-\frac{a}{\sqrt{1-a^2x^2}}\)

\(\frac{d}{dx}\arctan(ax)=\frac{a}{1+a^2x^2}\)

\(\frac{d}{dx}\cot^{-1}(ax)=-\frac{a}{1+a^2x^2}\)

\(\frac{d}{dx}\sec^{-1}(ax)=\frac{a}{|ax|\sqrt{a^2x^2-1}}\)

\(\frac{d}{dx}\csc^{-1}(ax)=-\frac{a}{|ax|\sqrt{a^2x^2-1}}\)

From Rogawski Calculus

\(\frac{d}{dx}[f(x)g(x)]=f(x)g^{'}(x)+g(x)f^{'}(x)\text{ 6. Product Rule}\)

\(\frac{d}{dx}\bigg[\frac{f(x)}{g(x)}\bigg]=\frac{g(x)f^{'}(x)-f(x)g^{'}(x)}{|g(x)|^2}\text{ 7. Quotient Rule}\)

\(\frac{d}{dx}f(g(x))=f^{'}(g(x))g^{'}(x)\text{ 8 Chain Rule}\)

\(\frac{d}{dx}\ln f(x)=\frac{f^{'}(x)}{f(x)}\text{ 12.}\)

\(\frac{d}{dx}(a^x)=(\ln a)a^x\text{ 26.}\)

\(\frac{d}{dx}(\log_ax)=\frac{1}{(\ln a)x}\text{ 28.}\)

\(\frac{d}{dx}(\sinh x)=\cosh(x)\text{ 29.}\)

\(\frac{d}{dx}(\cosh x)=\sinh x\text{ 30.}\)

\(\frac{d}{dx}(\tanh x)=sech^2x\text{ 31.}\)

\(\frac{d}{dx}(csch \,x)=-csch\,x\,\coth x\text{ 32.}\)

\(\frac{d}{dx}(sech\,x)=-sech\,x\tanh x\text{ 33.}\)

\(\frac{d}{dx}(\coth x)=-\ csch^2\,x\text{ 34.}\)

\(\frac{d}{dx}(\sinh^{-1}x)=\frac{1}{\sqrt{1+x^2}}\text{ 35.}\)

\(\frac{d}{dx}(\cosh^{-1}x)=\frac{1}{\sqrt{x^2-1}}\text{ 36.}\)

\(\frac{d}{dx}(\tanh^{-1}x)=\frac{1}{1-x^2}\text{ 37.}\)

\(\frac{d}{dx}(csch^{-1}\,x)=-\frac{1}{|x|\sqrt{x^2+1}}\text{ 38.}\)

\(\frac{d}{dx}(sech^{-1}\,x)=-\frac{1}{x\sqrt{1-x^2}}\text{ 39.}\)

\(\frac{d}{dx}(\coth^{-1} x)=\frac{1}{1-x^2}\text{ 40.}\)