This table collects the fundamental rules of differentiation.
Elementary Derivatives
| Function \( f(x) \) | Derivative \( f'(x) \) |
|---|---|
| \( c \) | \( 0 \) |
| \( x \) | \( 1 \) |
| \( x^n \) | \( nx^{n-1} \) |
| \( e^x \) | \( e^x \) |
| \( \ln(x) \) | \( \displaystyle\frac{1}{x} \) |
| \( \sqrt{x} \) | \( \displaystyle\frac{1}{2\sqrt{x}} \) |
| \( x^x \) | \( x^x (1 + \ln x) \) |
| \( |x| \) | \( \displaystyle\frac{x}{|x|} \), \( x \neq 0 \) |
| \( \ln|x| \) | \( \displaystyle\frac{1}{x} \), \( x \neq 0 \) |
| \( \text{sgn}(x) \) | \( 0 \), \( x \neq 0 \) |
Differentiation Rules
| Operation | Derivative |
|---|---|
| \( [f(x) \pm g(x)]' \) | \( f'(x) \pm g'(x) \) |
| \( [f(x) \cdot g(x)]' \) | \( f'(x)g(x) + f(x)g'(x) \) |
| \( \left[\displaystyle\frac{f(x)}{g(x)}\right]' \) | \( \displaystyle\frac{f'(x)g(x) - f(x)g'(x)}{g^2(x)} \) |
| \( [f(g(x))]' \) | \( f'(g(x)) \cdot g'(x) \) |
Trigonometric Functions
| \( f(x) \) | \( f'(x) \) |
|---|---|
| \( \sin(x) \) | \( \cos(x) \) |
| \( \cos(x) \) | \( -\sin(x) \) |
| \( \tan(x) \) | \( \sec^2(x) \) |
| \( \cot(x) \) | \( -\csc^2(x) \) |
| \( \sec(x) \) | \( \sec(x)\tan(x) \) |
| \( \csc(x) \) | \( -\csc(x)\cot(x) \) |
| \( \sin^2(x) \) | \( 2\sin(x)\cos(x) \) |
| \( \cos^2(x) \) | \( -2\cos(x)\sin(x) \) |
Inverse Trigonometric Functions
| \( f(x) \) | \( f'(x) \) |
|---|---|
| \( \arcsin(x) \) | \( \displaystyle\frac{1}{\sqrt{1 - x^2}} \) |
| \( \arccos(x) \) | \( \displaystyle -\frac{1}{\sqrt{1 - x^2}} \) |
| \( \arctan(x) \) | \( \displaystyle\frac{1}{1 + x^2} \) |
Hyperbolic Functions
| \( f(x) \) | \( f'(x) \) |
|---|---|
| \( \sinh(x) \) | \( \cosh(x) \) |
| \( \cosh(x) \) | \( \sinh(x) \) |
| \( \tanh(x) \) | \( \text{sech}^2(x) \) |
| \( \text{arsinh}(x) \) | \( \displaystyle\frac{1}{\sqrt{x^2 + 1}} \) |
| \( \text{arcosh}(x) \) | \( \displaystyle\frac{1}{\sqrt{x^2 - 1}} \) |
| \( \text{artanh}(x) \) | \( \displaystyle\frac{1}{1 - x^2} \) |
Exponentials and Logarithms
| \( f(x) \) | \( f'(x) \) |
|---|---|
| \( a^x \) | \( a^x \ln(a) \) |
| \( \log_a(x) \) | \( \displaystyle\frac{1}{x \ln a} \) |
Functions with Variable Exponents
| \( f(x) \) | \( f'(x) \) |
|---|---|
| \( x^\alpha \) | \( \alpha x^{\alpha - 1} \) |
| \( a^{g(x)} \) | \( a^{g(x)} \cdot \ln(a) \cdot g'(x) \) |
| \( \ln(g(x)) \) | \( \displaystyle\frac{g'(x)}{g(x)} \) |
| \( \ln|g(x)| \) | \( \displaystyle\frac{g'(x)}{g(x)} \) with \( g(x) \neq 0 \) |
Derivatives of Discontinuous or Piecewise Functions
| \( f(x) \) | \( f'(x) \) |
|---|---|
| \( \lfloor x \rfloor \) | Not differentiable at integer points |
| \( \lceil x \rceil \) | Not differentiable at integer points |
| \( |f(x)| \) | \( \displaystyle\frac{f(x)}{|f(x)|} \cdot f'(x) \), where defined |