# Table of Derivatives

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 ${\sinh{{\left({x}\right)}}}$ ${\cosh{{\left({x}\right)}}}$ ${\cosh{{\left({x}\right)}}}$ ${\sinh{{\left({x}\right)}}}$ ${\tanh{{\left({x}\right)}}}$ $\frac{{1}}{{{{\cosh}}^{{2}}{\left({x}\right)}}}={\text{sech}}^{{2}}{\left({x}\right)}$ ${\coth{{\left({x}\right)}}}$ $-\frac{{1}}{{{{\sinh}}^{{2}}{\left({x}\right)}}}=-{\operatorname{csch}}^{{2}}{\left({x}\right)}$ $\text{sech}{\left({x}\right)}=\frac{{1}}{{{\cosh{{\left({x}\right)}}}}}$ $-\text{sech}{\left({x}\right)}{\tanh{{\left({x}\right)}}}$ $\operatorname{csch}{\left({x}\right)}=\frac{{1}}{{{\sinh{{\left({x}\right)}}}}}$ $-\operatorname{csch}{\left({x}\right)}{\coth{{\left({x}\right)}}}$ 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 ${c}$ ${0}$ ${g{{\left({x}\right)}}}+{h}{\left({x}\right)}$ ${g{'}}{\left({x}\right)}+{h}'{\left({x}\right)}$ ${g{{\left({x}\right)}}}-{h}{\left({x}\right)}$ ${g{'}}{\left({x}\right)}-{h}'{\left({x}\right)}$ ${c}\cdot{g{{\left({x}\right)}}}$ ${c}\cdot{g{'}}{\left({x}\right)}$ ${g{{\left({x}\right)}}}{h}{\left({x}\right)}$ ${g{'}}{\left({x}\right)}{h}{\left({x}\right)}+{g{{\left({x}\right)}}}{h}'{\left({x}\right)}$ $\frac{{{g{{\left({x}\right)}}}}}{{{h}{\left({x}\right)}}}$ $\frac{{{g{'}}{\left({x}\right)}{h}{\left({x}\right)}-{g{{\left({x}\right)}}}{h}'{\left({x}\right)}}}{{{{h}}^{{2}}{\left({x}\right)}}}$ ${g{{\left({h}{\left({x}\right)}\right)}}}$ ${g{'}}{\left({h}{\left({x}\right)}\right)}\cdot{h}'{\left({x}\right)}$ ${{f}}^{{-{1}}}{\left({x}\right)}$ $\frac{{1}}{{{f{'}}{\left({{f}}^{{-{1}}}{\left({x}\right)}\right)}}}$