The Derivative Calculator finds the derivative of a function with respect to a chosen variable. A derivative describes the instantaneous rate of change of a function and is also the slope of the function's tangent line at a point where the derivative exists.
Derivative Formula
The derivative can be defined using the limit f′(x) = limh→0 [f(x+h) − f(x)] ÷ h. For many expressions, standard differentiation rules make the calculation much shorter. For example, the power rule gives d(xⁿ)/dx = nxⁿ⁻¹.
How to Use the Calculator
- Enter the function in the notation supported by the calculator.
- Choose the variable if more than one variable is possible.
- Calculate the derivative.
- Check the resulting expression and any stated domain restrictions.
Example
For f(x) = x³ + 4x, the derivative is f′(x) = 3x² + 4. At x = 2, the derivative is 16, which represents the tangent slope there.
Why Derivatives Are Useful
Derivatives are used to study increasing and decreasing behavior, rates of change, tangent lines, optimization, motion, and many models in science and engineering.