# Formulas and Rules for Derivative in Calculus



## Formulas

### $\dfrac{d f(x)}{dx}$

$x^n$ $n x^{n-1}$
$e^x$ $e^x$
$b^x$ $\ln b \cdot b^x$
$\ln x$ $\dfrac{1}{x}$
$\log_b x$ $\dfrac{1}{ x \ln b}$
$\sin x$ $\cos x$
$\cos x$ $- \sin x$
$\tan x$ $\sec^2 x$
$\cot x$ $- \csc^2 x$
$\sec x$ $\sec x \tan x$
$\csc x$ $- \csc x \cot x$
$\sin^{-1} x$ $\dfrac{1}{\sqrt{1-x^2}}$
$\cos^{-1} x$ $- \dfrac{1}{\sqrt{1-x^2}}$
$\tan^{-1} x$ $\dfrac{1}{1+x^2}$
$\sinh x$ $\cosh x$
$\cosh x$ $\sinh x$
$\tanh x$ $\text{sech}^2 x$
$\coth x$ $- \text{csch}^2 x$
$\text{sech} \; x$ $-\text{sech} \; x \tanh x$
$\text{csch} \; x$ $- \text{csch} \; x \coth x$
$\sinh^{-1} x$ $\dfrac{1}{\sqrt{x^2+1}}$
$\cosh^{-1} x$ $\dfrac{1}{\sqrt{x^2-1}}$
$\tanh^{-1} x$ $\dfrac{1}{1-x^2}$
$\coth^{-1} x$ $\dfrac{1}{1-x^2}$

## Rules

Let $u$ and $v$ be two functions with derivatives $u'$ and $v'$
• Linearity : $\dfrac{d(a u + b v)}{dx} = a u' + b v'$ , $a$ and $b$ are constants
• Derivative of Product : $\dfrac{d(u v)}{dx} = u' v + u v'$
• Derivative of Quotient : $\dfrac{d\left(\dfrac{u}{v}\right)}{dx} = \dfrac{u' v - u v'}{v^2}$
• Derivative of Composite : $\dfrac{d(u (v) )}{dx} = \dfrac{du}{dv} \dfrac{dv}{dx}$
• Derivative of a Function with a constant power: $\dfrac{d(u^n)}{dx} = n u'u^{n-1}$
• Derivative of a Function to the power of another function: $\dfrac{d(u^v)}{dx} = \left(v' ln (u) + \dfrac{u'}{u} v \right) u^v$