Differentiation Rules

$ \DeclareMathOperator{\arccot}{arccot} $

$ \left( {C{u}'} \right)=C{u}' $ Constant rule (C is a constant)

$ \left( {u+v} \right)^{\prime }={u}'+{v}' $ Addition rule

$ \left( {u-v} \right)^{\prime }={u}'-{v}' $ Subtraction rule

$ \left( {uv} \right)^{\prime }={u}'v+u{v}' $ Multiplication rule

$ \left( {\frac{u}{v}} \right)^{\prime }=\frac{{u}'v-u{v}'}{v^{2}} $ Division rule

$ \frac{du}{dx}=\frac{du}{dy}\frac{dy}{dx}$ Chain rule

$ \left( {x^{n}} \right)^{\prime }=nx^{n-1} $

$ \left( {e^{x}} \right)^{\prime }=e^{x} $

$ \left( {a^{x}} \right)^{\prime }=a^{x}\ln a $

$ \left( {\ln x} \right)^{\prime }=\frac{1}{x} $

$ \left( {\sin x} \right)^{\prime }=\cos x $

$ \left( {\cos x} \right)^{\prime }=-\sin x $

$ \left( {\tan x} \right)^{\prime }=\sec^{2}x $

$ \left( {\cot x} \right)^{\prime }=-\cos ec^{2}x $

$ \left( {\sinh x} \right)^{\prime }=\cosh x $

$ \left( {\cosh x} \right)^{\prime }=\sinh x $

$ \left( {\log_{a} x} \right)^{\prime }=\frac{1}{x}\log_{a} e $

$ \left( {\arcsin x} \right)^{\prime }=\frac{1}{\sqrt {1-x^{2}} } $

$ \left( {\arccos x} \right)^{\prime }=-\frac{1}{\sqrt {1-x^{2}} } $

$ \left( {\arctan x} \right)^{\prime }=\frac{1}{1+x^{2}} $

$ \left( {\arccot x} \right)^{\prime }=-\frac{1}{1+x^{2}} $