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Derivative Calculator

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Enter your polynomial's coefficients to find its derivative instantly, using the power rule.

Derivative Calculator

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f(x) = 3x³ + 5x² - 2x + 7
f'(x)
9x² + 10x - 2
Covers polynomials (terms with x raised to a whole-number power) - not trigonometric, logarithmic, or other non-polynomial functions.

The formula (power rule)

d/dx (axⁿ) = a × n × xⁿ⁻¹ Applied to each term, then combined
Example

f(x) = 3x³ + 5x² - 2x + 7: derivative of 3x³ is 9x², of 5x² is 10x, of -2x is -2, and the constant 7 disappears. f'(x) = 9x² + 10x - 2.

Step-by-step guide

  1. Enter the coefficient for each power of x in your polynomial, leaving unused terms at 0.
  2. Read the derivative, updated instantly as you type.

Why the power rule works the way it does

The power rule (bring the exponent down as a multiplier, then reduce the exponent by one) comes from the formal limit definition of a derivative, but its pattern becomes intuitive once you see it applied repeatedly: a constant term (x⁰) always vanishes, because a flat, unchanging value has zero rate of change - there's nothing to differentiate. Every other term follows the same mechanical rule regardless of degree, which is exactly what makes polynomial differentiation systematic rather than something that has to be worked out from scratch for each new function.

Common mistakes

Forgetting that a constant term's derivative is 0, not the constant itself - a flat value has no rate of change.
Applying the power rule to non-polynomial terms (like sin(x) or ln(x)) - those follow entirely different derivative rules not covered by this calculator.

Frequently asked questions

Does this handle trigonometric or logarithmic functions?

No - this covers polynomials only (terms with x raised to whole-number powers). Functions like sin(x), cos(x), eⁿ, or ln(x) follow different derivative rules not covered here.

What's the highest degree polynomial this handles?

Up to degree 5 (x⁵) - for lower-degree polynomials, simply leave the higher-power coefficient fields at 0.

Can I evaluate the derivative at a specific x value?

This tool shows the derivative expression itself - to evaluate it at a specific x, substitute that value into the resulting expression manually.

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