Derivative Calculator: Find the Derivative of a Function Online

Calculating complex derivatives can quickly become a mathematical challenge, especially when you need to maintain total confidentiality for your work. Our free derivative calculator solves this by using a 100% client-side technology, processing your equations directly in your browser without ever storing them on our servers. You get instant and accurate results, combining algorithmic rigor with absolute security. Boost your productivity with a powerful tool, designed to meet the needs of both students and the most demanding professionals.

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).

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