Polynomial Derivative Calculator
For a polynomial given by its coefficients — the derivative and the indefinite integral as new polynomials, the value and slope at a chosen x, the tangent line there, and the definite integral between two bounds.
Differentiating a polynomial is term by term: each aₙxⁿ becomes n·aₙxⁿ⁻¹.
How the polynomial derivative calculator works
Differentiating a polynomial is term by term: each aₙxⁿ becomes n·aₙxⁿ⁻¹. Integrating goes the other way: aₙxⁿ becomes aₙxⁿ⁺¹/(n+1). Evaluating at a point gives the value and the slope, which together fix the tangent line; evaluating the integral at two bounds and subtracting gives the signed area under the curve. Nothing here is approximate — the arithmetic is exact for any polynomial.
Formula: d/dx aₙxⁿ = n aₙ xⁿ⁻¹; ∫ aₙxⁿ dx = aₙ xⁿ⁺¹ / (n+1); tangent: y = f(x₀) + f′(x₀)(x − x₀)
Worked examples
| Inputs | Derivative f′(x) | Note |
|---|---|---|
| 2x³ − 3x² + 5 at x = 2 | 6x^2 − 6x | slope 12 |
| A parabola | 2x | area below the axis |
| A straight line | 3 | constant derivative |
FAQFrequently asked questions
How do I differentiate a polynomial?
Multiply each coefficient by its power and reduce the power by one; constants vanish. 2x³ − 3x² + 5 becomes 6x² − 6x.
How do I integrate one?
Raise each power by one and divide by the new power, then add a constant. The definite integral is the difference of that antiderivative at the two bounds — the signed area under the curve.
What does the slope at a point tell me?
The rate of change there and the direction of the tangent. Zero slope marks a maximum, minimum or saddle; the second derivative says which.
Why signed area?
Where the curve is below the axis the integral counts negative. Total enclosed area needs the integral split at the roots and the parts added without sign.
Can it handle other functions?
No — polynomials only, where the rules are exact and simple. Trigonometric, exponential and rational functions need their own rules or numerical methods.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.