Derivative Calculator — Free Step-by-Step Solutions for Any Function

A derivative calculator is a free online tool that computes the exact derivative — the instantaneous rate of change — of any function. Type a function such as x²·sin(x) or e^x, click Calculate, and instantly get the exact symbolic result with a complete step-by-step rule explanation, interactive graph, and critical point analysis. No sign-up required.

Yes — completely. Every calculation includes a full step-by-step solution showing exactly which rule was applied at each stage (power rule, chain rule, product rule, quotient rule, etc.) and the mathematical justification. Each step is numbered and colour-coded by rule type, so you can follow the working from start to finish. This makes our calculator ideal for students checking homework, teachers creating examples, and anyone who wants to understand why the answer is correct.

Yes, fully. Our differentiation calculator supports all six standard trig functions (sin, cos, tan, cot, sec, csc), all six inverse trig functions (arcsin, arccos, arctan, arccot, arcsec, arccsc), and all six hyperbolic functions (sinh, cosh, tanh, coth, sech, csch). Composite trig expressions like sin(x²), ln(cos(x)), and tan(e^x) are handled automatically via the chain rule.

Yes. Enter an equation such as x^2 + y^2 = 25 or x^3 + y^3 = 6*x*y directly into the input field. The calculator automatically detects the implicit form, differentiates both sides with respect to x (treating y as a function of x), collects all dy/dx terms, and solves for dy/dx — showing every step of the implicit differentiation process.

The derivative of sin(x) is cos(x). Written formally: d/dx[sin(x)] = cos(x). This is a fundamental trigonometric derivative result. Type sin(x) into the calculator above and click Calculate to see the full step-by-step working.

The derivative of e^x is e^x — the only function equal to its own derivative. For composites, the chain rule gives d/dx[e^(f(x))] = e^(f(x))·f'(x). Example: d/dx[e^(2x)] = 2e^(2x). Type e^(2*x) into the calculator to see the full chain rule working.

Yes. Select the derivative order — 2nd, 3rd, 4th, or 5th — from the Derivative Order dropdown before clicking Calculate. The engine applies differentiation repeatedly and shows every intermediate step for each order. The second derivative f''(x) is particularly useful for the Second Derivative Test: f''(c) < 0 indicates a local maximum; f''(c) > 0 indicates a local minimum; f''(c) = 0 requires further analysis. The third derivative (jerk) is used in physics and motion analysis.

Our derivative calculator supports all standard differentiation rules: Power Rule, Constant Rule, Sum and Difference Rule, Constant Multiple Rule, Product Rule, Quotient Rule, Chain Rule, Exponential Rule (e^x and a^x), Natural Logarithm Rule (d/dx[ln(x)] = 1/x), all trigonometric derivatives, all inverse trig derivatives, hyperbolic function derivatives, Logarithmic Differentiation (for functions like x^x), Implicit Differentiation, and Partial Differentiation for multivariable functions.

This derivative solver uses a dual symbolic Computer Algebra System (CAS) — Fast Mode for instant results and Accurate Mode (powered by a full Python-grade symbolic engine) for maximum precision. Both modes deliver exact symbolic derivatives, not rounded decimal approximations. Results like 2x·cos(x²) rather than imprecise floating-point numbers. This is the same class of symbolic engine used in professional mathematical software, making it one of the most reliable online derivative calculators available.

Yes — completely and permanently free with no sign-up, no subscription, and no limits on calculations. Every feature — step-by-step solutions, interactive graphs, higher-order derivatives (up to 5th), implicit differentiation, partial derivatives, integrals, and limits — is available at zero cost. Unlike Symbolab, which locks step-by-step solutions behind a paywall, or Wolfram Alpha Pro, which charges for full working, everything on DerivativeCalculus.com is free forever.