This table collects the main differentiation rules.
Basic Derivatives
| Function \( f(x) \) | Derivative \( f'(x) \) |
|---|---|
| \( c \) | \( 0 \) |
| \( x \) | \( 1 \) |
| \( x^n \) | \( nx^{n-1} \) |
| \( e^x \) | \( e^x \) |
| \( \ln(x) \) | \( \frac{1}{x} \) |
| \( \sqrt{x} \) | \( \frac{1}{2\sqrt{x}} \) |
| \( x^x \) | \( x^x(1+\ln x) \) |
| \( |x| \) | \( \frac{x}{|x|},\ x\neq0 \) |
| \( \ln|x| \) | \( \frac{1}{x},\ x\neq0 \) |
| \( \mathrm{sgn}(x) \) | \( 0,\ x\neq0 \) |
Differentiation Rules
| Operation | Derivative |
|---|---|
| \( [f(x)\pm g(x)]' \) | \( f'(x)\pm g'(x) \) |
| \( [f(x)\cdot g(x)]' \) | \( f'(x)g(x)+f(x)g'(x) \) |
| \( \left[\frac{f(x)}{g(x)}\right]' \) | \( \frac{f'(x)g(x)-f(x)g'(x)}{g^2(x)} \) |
| \( [f(g(x))]' \) | \( f'(g(x))\cdot g'(x) \) |
Trigonometric Functions
| Function \( f(x) \) | Derivative \( f'(x) \) |
|---|---|
| \( \sin(x) \) | \( \cos(x) \) |
| \( \cos(x) \) | \( -\sin(x) \) |
| \( \tan(x) \) | \( \sec^2(x) \) |
| \( \cot(x) \) | \( -\csc^2(x) \) |
| \( \sec(x) \) | \( \sec(x)\tan(x) \) |
| \( \csc(x) \) | \( -\csc(x)\cot(x) \) |
| \( \sin^2(x) \) | \( 2\sin(x)\cos(x) \) |
| \( \cos^2(x) \) | \( -2\cos(x)\sin(x) \) |
Inverse Trigonometric Functions
| Function \( f(x) \) | Derivative \( f'(x) \) |
|---|---|
| \( \arcsin(x) \) | \( \frac{1}{\sqrt{1-x^2}} \) |
| \( \arccos(x) \) | \( -\frac{1}{\sqrt{1-x^2}} \) |
| \( \arctan(x) \) | \( \frac{1}{1+x^2} \) |
Hyperbolic Functions
| Function \( f(x) \) | Derivative \( f'(x) \) |
|---|---|
| \( \sinh(x) \) | \( \cosh(x) \) |
| \( \cosh(x) \) | \( \sinh(x) \) |
| \( \tanh(x) \) | \( \mathrm{sech}^2(x) \) |
| \( \mathrm{arsinh}(x) \) | \( \frac{1}{\sqrt{x^2+1}} \) |
| \( \mathrm{arcosh}(x) \) | \( \frac{1}{\sqrt{x^2-1}} \) |
| \( \mathrm{artanh}(x) \) | \( \frac{1}{1-x^2} \) |
Exponentials and Logarithms
| Function \( f(x) \) | Derivative \( f'(x) \) |
|---|---|
| \( a^x \) | \( a^x\ln(a) \) |
| \( \log_a(x) \) | \( \frac{1}{x\ln(a)} \) |