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Calculus

Polynomial Derivative

Enter a polynomial in x using ^ for exponents. The calculator differentiates it exactly, applying the power rule d/dx(c·xⁿ) = c·n·xⁿ⁻¹ to each term and dropping constants. Every term is shown so you can follow the differentiation.

Polynomial Derivative

Differentiate a polynomial exactly with the power rule.

Try:
Answerf'(x) = 9x² − 10x + 2
  1. Functionf(x) = 3x³ − 5x² + 2x − 7
  2. Constant termd/dx(-7) = 0
  3. Power ruled/dx(2x) = 2
  4. Power ruled/dx(-5x²) = -10x
  5. Power ruled/dx(3x³) = 9x²
  6. Derivativef'(x) = 9x² − 10x + 2
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Formula and method

d/dx(c·xⁿ) = c·n·xⁿ⁻¹, d/dx(constant) = 0

Each term c·xⁿ is differentiated independently using the power rule: multiply the coefficient c by the exponent n to get the new coefficient, then reduce the exponent by one. Constant terms (n = 0) vanish. The resulting terms are collected and written as the derivative polynomial.

Worked examples

Find the derivative of x³ Apply the power rule to differentiate a single term. Find the derivative of 3x² − 5x + 2 Differentiate a quadratic polynomial term by term.

Learn why this works

Derivative Rules from First Principles Understand where the power rule d/dx xⁿ = nxⁿ⁻¹ comes from → Proof of the Product Rule Full derivation of (fg)′ = f′g + fg′ →

Key terms

Coefficient A constant multiplier in front of a variable or a term of a polynomial. Derivative The instantaneous rate of change of a function at a point.

Frequently asked questions

What is the power rule?

The derivative of c·xⁿ is c·n·xⁿ⁻¹. The derivative of a constant is 0.

How do I type a polynomial?

Use ^ for exponents and combine terms with + and −, for example 3x^3 - 5x^2 + 2x - 7.

Is the result exact?

Yes. Polynomial differentiation is symbolic and exact, not a numeric approximation.

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