Table of Derivatives

Related calculator: Online Derivative Calculator with Steps

Below is the list of the most common derivatives.

$$$f{{\left({x}\right)}}$$$ $$$f{'}\left({x}\right)$$$
Power Rule
$$$x^n$$$ $$$nx^{n-1}$$$
Exponential Function
$$$a^x$$$ $$${\ln{{\left({a}\right)}}}{{a}}^{{x}}$$$
$$${{e}}^{{x}}$$$ $$${{e}}^{{x}}$$$
Logarithmic Function
$$${\log}_{{a}}{\left({x}\right)}$$$ $$$\frac{{1}}{{{x}{\ln{{\left({a}\right)}}}}}$$$
$$${\ln}{\left|{x}\right|}$$$ $$$\frac{{1}}{{x}}$$$
Trigonometric Functions
$$${\sin{{\left({x}\right)}}}$$$ $$${\cos{{\left({x}\right)}}}$$$
$$${\cos{{\left({x}\right)}}}$$$ $$$-{\sin{{\left({x}\right)}}}$$$
$$${\tan{{\left({x}\right)}}}$$$ $$$\frac{{1}}{{{{\cos}}^{{2}}{\left({x}\right)}}}={{\sec}}^{{2}}{\left({x}\right)}$$$
$$${\cot{{\left({x}\right)}}}$$$ $$$-\frac{{1}}{{{{\sin}}^{{2}}{\left({x}\right)}}}=-{{\csc}}^{{2}}{\left({x}\right)}$$$
$$${\sec{{\left({x}\right)}}}=\frac{{1}}{{{\cos{{\left({x}\right)}}}}}$$$ $$${\sec{{\left({x}\right)}}}{\tan{{\left({x}\right)}}}$$$
$$${\csc{{\left({x}\right)}}}=\frac{{1}}{{{\sin{{\left({x}\right)}}}}}$$$ $$$-{\csc{{\left({x}\right)}}}{\cot{{\left({x}\right)}}}$$$
Inverse Trigonometric Functions
$$${\operatorname{arcsin}{{\left({x}\right)}}}$$$ $$$\frac{{1}}{{\sqrt{{{1}-{{x}}^{{2}}}}}}$$$
$$${\operatorname{arccos}{{\left({x}\right)}}}$$$ $$$-\frac{{1}}{{\sqrt{{{1}-{{x}}^{{2}}}}}}$$$
$$${\operatorname{arctan}{{\left({x}\right)}}}$$$ $$$\frac{{1}}{{{1}+{{x}}^{{2}}}}$$$
$$$\text{arccot}{\left({x}\right)}$$$ $$$-\frac{{1}}{{{1}+{{x}}^{{2}}}}$$$
$$$\text{arcsec}{\left({x}\right)}$$$ $$$\frac{{1}}{{{x}\sqrt{{{{x}}^{{2}}-{1}}}}}$$$
$$$\text{arccsc}{\left({x}\right)}$$$ $$$-\frac{{1}}{{{x}\sqrt{{{{x}}^{{2}}-{1}}}}}$$$
Hyperbolic Functions
$$${\sinh{{\left({x}\right)}}}$$$ $$${\cosh{{\left({x}\right)}}}$$$
$$${\cosh{{\left({x}\right)}}}$$$ $$${\sinh{{\left({x}\right)}}}$$$
$$${\tanh{{\left({x}\right)}}}$$$ $$$\frac{{1}}{{{{\cosh}}^{{2}}{\left({x}\right)}}}={\text{sech}}^{{2}}{\left({x}\right)}$$$
$$${\coth{{\left({x}\right)}}}$$$ $$$-\frac{{1}}{{{{\sinh}}^{{2}}{\left({x}\right)}}}=-{\text{csch}}^{{2}}{\left({x}\right)}$$$
$$$\text{sech}{\left({x}\right)}=\frac{{1}}{{{\cosh{{\left({x}\right)}}}}}$$$ $$$-\text{sech}{\left({x}\right)}{\tanh{{\left({x}\right)}}}$$$
$$$\text{csch}{\left({x}\right)}=\frac{{1}}{{{\sinh{{\left({x}\right)}}}}}$$$ $$$-\text{csch}{\left({x}\right)}{\coth{{\left({x}\right)}}}$$$
Inverse Hyperbolic Functions
$$$\text{arcsinh}{\left({x}\right)}$$$ $$$\frac{{1}}{{\sqrt{{{{x}}^{{2}}+{1}}}}}$$$
$$$\text{arccosh}{\left({x}\right)}$$$ $$$\frac{{1}}{{\sqrt{{{{x}}^{{2}}-{1}}}}}$$$
$$$\text{arctanh}{\left({x}\right)}$$$ $$$\frac{{1}}{{{1}-{{x}}^{{2}}}}$$$
$$$\text{arccot}\text{h}{\left({x}\right)}$$$ $$$\frac{{1}}{{{1}-{{x}}^{{2}}}}$$$
$$$\text{arcsec}\text{h}{\left({x}\right)}$$$ $$$-\frac{{1}}{{{x}\sqrt{{{1}-{{x}}^{{2}}}}}}$$$
$$$\text{arccsc}\text{h}{\left({x}\right)}$$$ $$$-\frac{{1}}{{{\left|{x}\right|}\sqrt{{{1}+{{x}}^{{2}}}}}}$$$
Differentiation Rules
$$${c}$$$ $$${0}$$$
$$${g{{\left({x}\right)}}}+{h}{\left({x}\right)}$$$ $$${g{'}}{\left({x}\right)}+{h}'{\left({x}\right)}$$$
$$${g{{\left({x}\right)}}}-{h}{\left({x}\right)}$$$ $$${g{'}}{\left({x}\right)}-{h}'{\left({x}\right)}$$$
$$${c}\cdot{g{{\left({x}\right)}}}$$$ $$${c}\cdot{g{'}}{\left({x}\right)}$$$
$$${g{{\left({x}\right)}}}{h}{\left({x}\right)}$$$ $$${g{'}}{\left({x}\right)}{h}{\left({x}\right)}+{g{{\left({x}\right)}}}{h}'{\left({x}\right)}$$$
$$$\frac{{{g{{\left({x}\right)}}}}}{{{h}{\left({x}\right)}}}$$$ $$$\frac{{{g{'}}{\left({x}\right)}{h}{\left({x}\right)}-{g{{\left({x}\right)}}}{h}'{\left({x}\right)}}}{{{{h}}^{{2}}{\left({x}\right)}}}$$$
$$${g{{\left({h}{\left({x}\right)}\right)}}}$$$ $$${g{'}}{\left({h}{\left({x}\right)}\right)}\cdot{h}'{\left({x}\right)}$$$
$$${{f}}^{{-{1}}}{\left({x}\right)}$$$ $$$\frac{{1}}{{{f{'}}{\left({{f}}^{{-{1}}}{\left({x}\right)}\right)}}}$$$