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Derivative Calculator

This derivative calculator needs no sign-up: enter a function of x and it returns the simplified derivative together with the steps and the rules used. On request you also get the second and third derivative plus the function value, slope and tangent line at a chosen point.

Source: Differentiation rules: sum, constant factor, product, power and chain rule, plus derivatives of sin, cos, tan, exp, ln (standard calculus material). Updated: .

e.g. x^3 - 4x + 1, sin(x)*x^2, e^(2x), ln(x)/x, sqrt(x). Allowed: + − * / ^, brackets, sin cos tan exp ln sqrt, e, pi
for function value, slope and tangent line; leave empty if not needed
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Graph of f and f′-4-3-2-1123456-20-10102030
f(x)f′(x)Tangent at x₀

Result

Derivative
x^2·cos(x) + 2x·sin(x)
All orders
  • f′(x) = x^2·cos(x) + 2x·sin(x)
Steps (first derivative)
  • Product rule (u·v)′ = u′·v + u·v′: (x^2·sin(x))′ = x^2·cos(x) + 2x·sin(x)
  • › Power rule (xⁿ)′ = n·xⁿ⁻¹: (x^2)′ = 2x
  • › Chain rule: outer derivative times inner derivative: (sin(x))′ = cos(x)
Function value and slope
f(1) = 0.841471; f′(1) = 2.22324
Tangent line at x₀
y = 2.22324x − 1.38177
Points where f′(x) = 0 (−10 to 10)
  • x = −8.09616: local minimum (f″ > 0), f(x) = −63.635
  • x = −5.08699: local maximum (f″ < 0), f(x) = 24.083
  • x = −2.28893: local minimum (f″ > 0), f(x) = −3.9453
  • x = 0: horizontal tangent, f″ = 0 – check the sign change of f′, f(x) = 0
  • x = 2.28893: local maximum (f″ < 0), f(x) = 3.9453
  • x = 5.08699: local minimum (f″ > 0), f(x) = −24.083
  • x = 8.09616: local maximum (f″ < 0), f(x) = 63.635
The result is simplified exactly but may look different from your textbook answer without being wrong. Derivatives hold on the domain of the function, e.g. x > 0 for ln(x).

How it is calculated

The derivative of a function tells you how fast its value changes: at every point it is the slope of the tangent line. This tool differentiates functions of one variable symbolically, simplifies the result and shows which rule was applied to which part.

The main differentiation rules

Higher derivatives, slope and tangent line

The second derivative describes curvature, the third the change of curvature. If you enter a point x₀, the tool computes the function value, the slope f′(x₀) and the tangent line y = f′(x₀)·(x − x₀) + f(x₀). The core is checked in its tests against the difference quotient. You can see the shape of the function in the function plotter; for roots and powers use the exponent calculator and the quadratic equation solver.

Graph, extrema and cross-check

Below the fields the tool draws f (solid) and f′ (dashed) in one picture. Enter a point x₀ and the point of contact and the tangent line are added. You can see directly: where f′ crosses the x-axis, f has a horizontal tangent; where f′ is positive, f rises.

The line “Points where f′(x) = 0” lists these points between −10 and 10 and classifies them with the second derivative: f″ < 0 is a local maximum, f″ > 0 a local minimum. If f″ = 0 as well, only the sign change of f′ decides – the tool says so. The points are found numerically and rounded to 6 significant digits.

Before anything is shown, every derivative is compared in the background with the difference quotient at 24 points. If something does not match, you get a message instead of a result. Derivatives are now available up to the fifth order. Not supported are arcsin, arccos, arctan, sinh, cosh, absolute values and several variables – you get a clear message for those. For the reverse operation, the antiderivative, use the integral calculator.

Frequently asked questions

How do I calculate a derivative?

Split the function into parts and apply the matching rule to each: the power rule for xⁿ, sum and constant factor rules for sums and numbers in front, the product or chain rule for combinations. The tool shows exactly these steps.

Which functions are supported?

Sums, products, fractions and powers of x as well as sin, cos, tan, exp, ln and roots, plus the constants e and pi. Expressions such as 2^x or x^x work too. Several variables and integrals are not included.

How do I type fractions and roots?

Fractions with a slash, such as 1/x or (x+1)/(x-2). Roots with sqrt(x) or as a power x^(1/2). Decimal numbers work with a point or a comma: 0.5x^2.

What is the tangent line at a point?

The straight line that touches the graph at x₀ and has the same slope there. It is y = f′(x₀)·(x − x₀) + f(x₀). The tool fills in the numbers for you.

Why does my result look different from the textbook answer?

Equal derivatives can be written in different ways, e.g. 2sin(x)·cos(x) and sin(2x). To check, plug in the same value: if both forms give the same number, they are equivalent.

Sources and legal basis

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