# 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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