Reviewed by CalcMulti Editorial Team·Last updated: ·← Calculus Hub
A derivative measures the instantaneous rate of change of a function — how fast the output changes as the input changes at any given point. Derivatives are the foundation of differential calculus and appear throughout physics, engineering, economics, and data science.
The derivative of f(x) is written f'(x) or df/dx and is defined as the limit: f'(x) = lim[h→0] (f(x+h) − f(x)) / h. This limit gives the slope of the tangent line to the curve y = f(x) at every point x.
Common rules make finding derivatives systematic: the power rule (d/dx[xⁿ] = nxⁿ⁻¹), the chain rule for composite functions, the product rule, and the quotient rule. This calculator applies all standard differentiation rules and shows the step-by-step working.
Worked example: if f(x) = 3x⁴ − 5x² + 7x − 2, then f'(x) = 12x³ − 10x + 7 by the power rule applied to each term.
f'(x) = lim[h→0] (f(x+h) − f(x)) / h
Common Derivatives
This calculator is for educational purposes only and does not constitute professional advice. Results are based on standard mathematical formulas. Always verify critical calculations with a qualified professional before making important decisions.