Derivative Calculator
Type a function and this calculator differentiates it instantly, step by step, for any variable.
Use ^ for powers, e.g. x^3 + 2x or sin(x)*x^2. Supports trig, exponentials and logs.
How to find a derivative by hand
- Break the function into terms added or subtracted together, and differentiate each one separately.
- For a power term xⁿ, multiply by the exponent and reduce the exponent by 1 (the power rule).
- For a product of two functions, use the product rule: (fg)' = f'g + fg'.
- For a function inside a function, use the chain rule: multiply the outer derivative by the inner derivative.
Worked example
Differentiate f(x) = x³ + 2x. By the power rule, the derivative of x³ is 3x², and the derivative of 2x is 2. Adding them gives f'(x) = 3x² + 2.
See every formula worked through on its own page in the maths formula list, or simplify an expression first in the Simplify Calculator.
Frequently asked questions
What is a derivative in calculus?
The derivative of a function measures how fast its output changes as its input changes — its instantaneous rate of change, or the slope of its graph at a point.
What is the power rule for derivatives?
If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹. Multiply by the exponent, then reduce the exponent by 1: the derivative of x³ is 3x².
What is the derivative of a constant?
Zero. A constant never changes, so its rate of change is always 0.
What is the derivative of sin(x)?
cos(x). More generally, the derivative of sin(kx) is k·cos(kx) by the chain rule.
