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 | |
| $$${c}$$$ | $$${0}$$$ |
| $$${g{{\left({x}\right)}}}+{h}{\left({x}\right)}$$$ | $$${g{'}}{\left({x}\right)}+{h}'{\left({x}\right)}$$$ |
| $$${g{{\left({x}\right)}}}-{h}{\left({x}\right)}$$$ | $$${g{'}}{\left({x}\right)}-{h}'{\left({x}\right)}$$$ |
| $$${c}\cdot{g{{\left({x}\right)}}}$$$ | $$${c}\cdot{g{'}}{\left({x}\right)}$$$ |
| $$${g{{\left({x}\right)}}}{h}{\left({x}\right)}$$$ | $$${g{'}}{\left({x}\right)}{h}{\left({x}\right)}+{g{{\left({x}\right)}}}{h}'{\left({x}\right)}$$$ |
| $$$\frac{{{g{{\left({x}\right)}}}}}{{{h}{\left({x}\right)}}}$$$ | $$$\frac{{{g{'}}{\left({x}\right)}{h}{\left({x}\right)}-{g{{\left({x}\right)}}}{h}'{\left({x}\right)}}}{{{{h}}^{{2}}{\left({x}\right)}}}$$$ |
| $$${g{{\left({h}{\left({x}\right)}\right)}}}$$$ | $$${g{'}}{\left({h}{\left({x}\right)}\right)}\cdot{h}'{\left({x}\right)}$$$ |
| $$${{f}}^{{-{1}}}{\left({x}\right)}$$$ | $$$\frac{{1}}{{{f{'}}{\left({{f}}^{{-{1}}}{\left({x}\right)}\right)}}}$$$ |