Derivative formulas
Reference of derivatives of basic functions (power, sine, exponential, logarithm) with a live differentiator.
| function f(x) | derivative f′(x) |
|---|---|
| c | 0 the derivative of a constant is zero |
| x | 1 |
| xⁿ | n·xⁿ⁻¹ power rule – exponent moves down, decreases by one |
| √x | 1 / (2√x) |
| 1/x | −1/x² |
| eˣ | eˣ the exponential is its own derivative |
| aˣ | aˣ·ln a |
| ln x | 1/x |
| log x | 1 / (x·ln 10) |
| sin x | cos x |
| cos x | −sin x |
| tan x | 1/cos²x |
| arcsin x | 1/√(1−x²) |
| arctan x | 1/(1+x²) |
How to use the formulas
Most functions are built from a few basic ones. If you differentiate each part using the table above and combine them with the differentiation rules (term-by-term sum, product, quotient, chain rule), you can differentiate even complex expressions. The most common is the power rule (xⁿ)′ = n·xⁿ⁻¹ – the exponent moves to the front as a factor and decreases by one.
What derivatives are used for
The derivative describes the rate of change and the slope of a graph. It's used in physics (velocity, acceleration), for finding maxima and minima, in economics, and in machine learning. See it visually on the derivative at a point page.
Tip: type any function into the field above to get its derivative symbolically.