Derivative Calculator: Find the Derivative of a Function Online

Calculate the symbolic derivative of a function, nth derivative and numerical value f'(a) with graph plot.

Function f(x)

f(x) =

x² x³−2x+1 sin(x) cos(x) eˣ ln(x) √x x²·sin(x) sin(x)/x (x²+1)/(x−1) x·eˣ arctan(x)

↵ Enter to differentiate

Enter a function and click Differentiate.

How to calculate a derivative online?

The derivative of a function f(x) represents the instantaneous rate of change of the function at each point. Written f'(x) or df/dx, it is fundamental in mathematical analysis, physics and economics. Our calculator uses symbolic differentiation to produce the exact derivative expression.

Main differentiation rules

The fundamental rules are: sum rule (f+g)' = f'+g', product rule (f·g)' = f'·g + f·g', quotient rule (f/g)' = (f'g−fg')/g², and chain rule (f∘g)' = f'(g(x))·g'(x). Standard derivatives: (xⁿ)' = n·xⁿ⁻¹, (sin x)' = cos x, (cos x)' = −sin x, (eˣ)' = eˣ, (ln x)' = 1/x.

Second derivative and interpretation

The second derivative f''(x) gives the curvature of the function. If f''(x) > 0, the function is convex (curving upward); if f''(x) < 0, it is concave. Inflection points are at zeros of f''(x). The second derivative is also acceleration in physics (derivative of velocity).

Tools in the same category

Statistics Calculator System of Equations Solver GCD and LCM Calculator Fraction Calculator Multiplication Tables Math Solver