Table of Derivative Rules & Formulas

This table provides essential derivative rules, including power, product, quotient, chain, and trigonometric differentiation formulas for quick reference.

Neetesh Kumar | February 06, 2025 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

Table of Content:

1. Derivative Formulas
2. Derivative Rules

Derivative Formulas:

$f(x)$	$f'(x)$
$Constant$	$0$
$x^n$	$n \cdot x^{n-1}$
$e^x$	$e^x$
$a^x$	$a^x .\ln a$
$\ln x$	$\dfrac{1}{x}$
$\log_a x$	$\dfrac{1}{x. \ln a}$
$abs(x)$	$\dfrac{x}{abs(x)} \text{ (for } x \neq 0 \text{)}$
$\sqrt{x}$	$\dfrac{1}{2.\sqrt{x}}$
$\sqrt[n]{x}$	$\dfrac{1}{n}. x^{(\frac{1}{n} - 1)}$
$\sin (x)$	$\cos (x)$
$\cos (x)$	$-\sin (x)$
$\tan (x)$	$\sec^2 (x)$
$\cot (x)$	$-\csc^2 (x)$
$\sec (x)$	$\sec (x) .\tan (x)$
$\csc (x)$	$-\csc (x) .\cot (x)$
$\sin^{-1} x$	$\dfrac{1}{\sqrt{1-x^2}}$
$\cos^{-1} x$	$-\dfrac{1}{\sqrt{1-x^2}}$
$\tan^{-1} x$	$\dfrac{1}{1+x^2}$
$\cot^{-1} x$	$-\dfrac{1}{1+x^2}$
$\sec^{-1} x$	$\dfrac{1}{abs(x). \sqrt{x^2 - 1}}$
$\csc^{-1} x$	$-\dfrac{1}{abs(x). \sqrt{x^2 - 1}}$
$\sinh (x)$	$\cosh (x)$
$\cosh (x)$	$\sinh (x)$
$\tanh (x)$	$\text{sech}^2 (x)$
$\coth (x)$	$-\text{csch}^2 (x)$
$\text{sech} (x)$	$-\text{sech} (x). \tanh (x)$
$\text{csch} (x)$	$-\text{csch} (x). \coth (x)$
$\sinh^{-1} (x)$	$\dfrac{1}{\sqrt{x^2 + 1}}$
$\cosh^{-1} (x)$	$\dfrac{1}{\sqrt{x^2 - 1}}$
$\tanh^{-1} (x)$	$\dfrac{1}{1 - x^2}$
$\coth^{-1} (x)$	$\dfrac{1}{1 - x^2}$
$\text{sech}^{-1} (x)$	$-\dfrac{1}{x. \sqrt{1 - x^2}}$
$\text{csch}^{-1} (x)$	$-\dfrac{1}{abs(x). \sqrt{1 + x^2}}$

Derivative Rules:

$\mathrm{Rule} \space \mathrm{Name}$	$\mathrm{Formula} \space \mathrm{Used}$
$\mathrm{Constant \space Rule}$	$\dfrac{d(c)}{dx} = 0$
$\mathrm{Sum \space Rule}$	$\dfrac{d}{dx}\bigg[f(x) + g(x)\bigg] = f'(x) + g'(x)$
$\mathrm{Difference \space Rule}$	$\dfrac{d}{dx}\bigg[f(x) - g(x)\bigg] = f'(x) - g'(x)$
$\mathrm{Power \space Rule}$	$\dfrac{d}{dx}\bigg[x^n\bigg] = n.x^{n-1}$
$\mathrm{Product \space Rule}$	$\dfrac{d}{dx}\bigg[f(x).g(x)\bigg] = f'(x).g(x) + f(x).g'(x)$
$\mathrm{Quotient \space Rule}$	$\dfrac{d}{dx}\bigg[\dfrac{f(x)}{g(x)}\bigg] = \dfrac{f'(x).g(x) - f(x).g'(x)}{\big[g(x)\big]^2}$
$\mathrm{Reciprocal \space Rule}$	$\dfrac{d}{dx}\left[\dfrac{1}{f(x)}\right] = -\dfrac{f'(x)}{\big[f(x)\big]^2}$
$\mathrm{Inverse \space Function \space Rule}$	$\dfrac{d}{dx}\bigg[f^{-1}(x)\bigg] = \dfrac{1}{f'\big(f^{-1}(x)\big)}$
$\mathrm{Composite \space Function \space Rule}$	$\dfrac{d}{dx}\bigg[f\bigg(g(x)\bigg)\bigg] = f'\bigg(g(x)\bigg) \cdot g'(x)$