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Calculator Answers

Derivative Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A derivative calculator is a mathematical utility that computes the rate of change or slope of a function with respect to an independent variable, a process known as differentiation. It applies standard calculus differentiation rules, such as the power, product, quotient, and chain rules, to determine the derivative function. In basic algebraic and polynomial settings, it simplifies the differentiation term by term. Students, physicists, and engineers use it to solve rate-of-change problems, find tangents, model physical velocities, and analyze the behavior and critical points of mathematical functions.

Type a polynomial in x and we'll differentiate it term by term using the power rule, with a clear breakdown of each step. Polynomials only — no trig or exponentials.

See also: Laplace Transform Calculator, Integral Calculator, and Taylor Series Calculator.

Quick Answer

Find the derivative of any polynomial function. Enter your function to see the term-by-term differentiation using the power rule.

Polynomials in x only — terms like 3x^2, -2x, and 5. No functions like sin or log.

Optional — leave blank or enter any real number.

Term-by-term steps

  • d/dx[3x^2] = 6x · (×2, exp − 1)
  • d/dx[2x] = 2 · (×1, exp − 1)
  • d/dx[1] = 0 (constant)
Derivative

f′(x) =

6x +2

Power rule applied term by term

f(x) (parsed)3x^2 +2x +1
Number of terms in f′2
f′(2)14

Polynomial derivatives only — power rule, term by term. For chain, product, quotient, or trig/log/exp derivatives, use a full CAS like Wolfram Alpha.

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Examples

f(x) = 3x² + 2x + 1

f′(x) = 6x + 2

f(x) = x³

f′(x) = 3x²

f(x) = 5x⁴ − 2x² + 7

f′(x) = 20x³ − 4x

f(x) = 4x

f′(x) = 4

How it works

Every polynomial term has the form c·xⁿ. The power rule says its derivative is the coefficient times the exponent, with the exponent dropped by one.

power rule · d/dx[c·xⁿ] = c·n·x^(n−1)

We apply this to each term and add the results. Constants (terms with no x) drop out because their derivative is zero.

Supported: polynomials in x · ^ for exponents · constants · negative coefficients. Unsupported: sin, cos, ln, e^x, products, quotients, chain rule.

Polynomials only. For trig, exponential, logarithmic, or composite functions, use a full computer-algebra system. We deliberately keep the scope narrow so every step is auditable.

For the limit-based definition that motivates the derivative, see the limit calculator.

Related Calculators

More tools from Education

Laplace Transform CalculatorFind the Laplace transform of common functions of time using a standard lookup table of transform pairs.Integral CalculatorCompute definite integrals of polynomials with the power rule, or numerically with Simpson's rule for safe expressions.Taylor Series CalculatorApproximate any differentiable function f(x) with a Taylor polynomial centered at a specified point a with successive derivatives.Partial Derivative CalculatorCalculate the partial derivative of any multi-variable function with respect to a chosen variable with optional point evaluation.

Frequently asked questions

Polynomials in x only — terms like 3x^2, -2x, and constants. The calculator does not handle trig, exponential, logarithmic, product/quotient, chain rule, or implicit differentiation. For anything beyond polynomial differentiation, use a full computer algebra system like Wolfram Alpha or SymPy.

For any term c·xⁿ, the derivative is c·n·x^(n−1). Multiply the coefficient by the exponent, then drop the exponent by one. Constants (terms with no x) differentiate to 0. We apply this rule to each term in your polynomial and sum the results.

Use ^ for exponents. Spaces are optional. Examples: 3x^2 + 2x + 1, x^3 - 4x, 5, -2x + 7, 0.5x^2. The variable is always x. You can use unicode minus (−) or ASCII minus (-).

Building a safe symbolic differentiation engine for those functions would require a real computer-algebra parser. We've intentionally kept the scope to polynomials so the math stays auditable and the formula reads as exactly the power rule. If you need a CAS, Wolfram Alpha and SymPy both handle the full toolkit.