Polynomial Derivative Calculator (Power Rule)
Instantly differentiate any polynomial using the Power Rule. Enter coefficients from highest to lowest degree — get the exact derivative expression, step-by-step logic, and a table of worked examples.
- Data verified · June 2026
- Edited by Martín Rodríguez
- Formula verified by automated tests
- Private — runs on your device
See step-by-step calculation
See this calculation step by step
See period-by-period detail
How to use this calculator
Follow this tool’s steps, then review its formula, assumptions, and limits below.
3,2,1; for x⁴−5x²+2 enter 1,0,−5,0,2. Zero-coefficient terms for missing powers must be included explicitly. Used in calculus courses, physics kinematics, economics optimization, and engineering analysis.When to use this calculator
- Finding instantaneous velocity v(t) = s′(t) from a polynomial position function — e.g., s(t)=4t³−2t²+t gives v(t)=12t²−4t+1.
- Locating critical points of a polynomial profit or cost function: set p′(x)=0 and solve for x to find maxima or minima.
- Computing the slope of a polynomial curve at a specific point for tangent-line equations in analytic geometry.
- Differentiating polynomial equations of motion in physics — e.g., h(t)=−16t²+80t+6 (height in feet) gives h′(t)=−32t+80 to find when vertical velocity equals zero at peak height.
Derivative ↔ integral: how the same term transforms under each operation
Differentiation and integration are inverse operations, and the Power Rule runs in both directions. Differentiating multiplies by the exponent and drops it by 1 (d/dx[axⁿ] = n·a·xⁿ⁻¹); integrating raises the exponent by 1 and divides by the new one (∫axⁿ dx = a·xⁿ⁺¹/(n+1) + C). Reading the derivative column and the integral column side by side shows why integrating a derivative returns the original (up to the constant C).
| Term | Derivative d/dx | Integral ∫ … dx | Power-rule effect |
|---|---|---|---|
| 6 (constant) | 0 | 6x + C | constant → vanishes / → linear |
| 5x | 5 | (5/2)x² + C | exponent 1 ↔ 0 / 1 ↔ 2 |
| x² | 2x | x³/3 + C | ×2, drop / ÷3, raise |
| 3x² | 6x | x³ + C | ×2 / ÷3 |
| x³ | 3x² | x⁴/4 + C | ×3, drop / ÷4, raise |
| x⁴ | 4x³ | x⁵/5 + C | ×4, drop / ÷5, raise |
| axⁿ (general) | n·a·xⁿ⁻¹ | a·xⁿ⁺¹/(n+1) + C | the two Power Rules, n ≠ −1 |
Sanity check any answer by reversing it: the integral of a polynomial's derivative reproduces the original polynomial except for the constant term (which differentiation erased). Constants always differentiate to 0, so every antiderivative carries +C.
Successive derivatives of p(x) = x⁴ (how degree falls by 1 each time)
Each differentiation lowers the polynomial degree by exactly 1, so a degree-n polynomial vanishes after n+1 derivatives. Below is x⁴ differentiated repeatedly — useful for the Second Derivative Test and Taylor-series coefficients.
| Order | Notation | Result | Degree |
|---|---|---|---|
| 0th (original) | p(x) | x⁴ | 4 |
| 1st | p′(x) | 4x³ | 3 |
| 2nd | p″(x) | 12x² | 2 |
| 3rd | p‴(x) | 24x | 1 |
| 4th | p⁗(x) | 24 | 0 |
| 5th | p⁽⁵⁾(x) | 0 | — (zero polynomial) |
To get second or higher derivatives in this calculator, feed the output's coefficients back in: x⁴ → 1,0,0,0,0 → 4x³ → 4,0,0,0 → 12x² → 12,0,0 → 24x, and so on.
How it works
How the Polynomial Derivative Is Calculated
The Power Rule is the single rule that differentiates every polynomial:
d/dx [a·xⁿ] = n·a·xⁿ⁻¹
For a full polynomial:
p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀
p′(x) = n·aₙxⁿ⁻¹ + (n−1)·aₙ₋₁xⁿ⁻² + … + 1·a₁Algorithm:
1. Assign degree d = (number of coefficients − 1).
2. For each coefficient aᵢ (position i, 0-indexed from left): new_coeff = aᵢ × (d−i); new_exponent = (d−i) − 1.
3. Drop the last term — the constant's derivative is always 0.
4. Assemble the result from highest to lowest remaining power.
---
Reference Table: Common Polynomial Derivatives
| Polynomial p(x) | Coefficients input | Derivative p′(x) |
|---|---|---|
| x² + 2x + 1 | 1, 2, 1 | 2x + 2 |
| 3x² + 2x + 1 | 3, 2, 1 | 6x + 2 |
| x³ − 3x + 5 | 1, 0, −3, 5 | 3x² − 3 |
| −16t² + 80t + 6 | −16, 80, 6 | −32t + 80 |
| x⁴ − 5x² + 2 | 1, 0, −5, 0, 2 | 4x³ − 10x |
| 2x⁵ − 4x³ | 2, 0, −4, 0, 0, 0 | 10x⁴ − 12x² |
| 4x³ − 9x | 4, 0, −9, 0 | 12x² − 9 |
| 7x + 4 | 7, 4 | 7 |
| 6 (constant) | 6 | 0 |
| x³ − 3x² + 3x − 1 | 1, −3, 3, −1 | 3x² − 6x + 3 |
---
Applied Examples
Physics — Projectile Height
Height in feet:
h(t) = −16t² + 80t + 6 → coefficients −16, 80, 6Velocity: h′(t) = −32t + 80
Set h′(t)=0 → t = 2.5 s (peak). Substituting: h(2.5) = 106 feet.
Economics — Profit Maximization
Profit in $000s:
P(q) = −2q³ + 12q² − 18q + 5 → coefficients −2, 12, −18, 5Marginal profit: P′(q) = −6q² + 24q − 18
Set P′(q)=0 → q=1 (local min) or q=3 (local max).
Geometry — Tangent Line
f(x) = x³ − 2x + 1 at x=2 → coefficients 1, 0, −2, 1f′(x) = 3x² − 2; slope at x=2: f′(2) = 10. Point: f(2) = 5.
Tangent line: y = 10x − 15.
---
Common Errors
1. Missing zero-coefficient terms. For 4x³ − 9x, enter 4, 0, −9, 0 — not 4, −9, 0. Skipping the x² term shifts every coefficient to the wrong degree.
2. Reversing coefficient order. Input must be highest-to-lowest degree. 1, 2, 3 means x²+2x+3 (derivative: 2x+2), not 3x²+2x+1.
3. Confusing derivative with integral. The Power Rule for derivatives reduces exponents by 1 and multiplies. Integration does the opposite — increases exponents and divides.
4. Constant term stays constant (wrong). d/dx[7] = 0, not 7. Every constant vanishes under differentiation.
5. Off-by-one degree. Three coefficients a, b, c = degree 2 (ax²+bx+c), not degree 3. Degree always equals len(coefficients) − 1.
Worked Example
3, 2, 1Frequently asked questions
What is the Power Rule for polynomial derivatives?
How do I enter coefficients for a polynomial with missing terms?
4, 0, −9, 0. Skipping zeros shifts every remaining coefficient to the wrong degree — 4, −9, 0 would be interpreted as 4x²−9x+0, giving the wrong derivative 8x−9 instead of the correct 12x²−9.What happens to the constant term when differentiating a polynomial?
What is the derivative of a constant polynomial like f(x) = 7?
7 (a single coefficient) and the calculator returns 0. Using the Power Rule: d/dx[7] = d/dx[7·x⁰] = 0·7·x⁻¹ = 0.How can I compute second or higher-order derivatives?
1,0,0,0,0 → p′(x)=4x³ → 4,0,0,0 → p″(x)=12x² → 12,0,0 → p‴(x)=24x → p⁽⁴⁾(x)=24. Second derivatives are used in the Second Derivative Test to classify critical points.How is polynomial differentiation used in kinematics?
Does the Power Rule apply to negative or fractional exponents?
Why does differentiating always reduce the polynomial's degree by exactly 1?
Sources & references
Methodology & trust
Math calculator with its formula verified automatically against MIT OpenCourseWare — Single Variable Calculus 18.01SC, per our editorial policy and methodology.
Updated: June 2026. Parameters are verified periodically against the cited sources.
Calculations run 100% in your browser. We do not store or transmit your data.
Indicative results. For critical decisions, consult a professional.
📌 How to cite this calculator
Rodríguez, M. (2026). Polynomial Derivative Calculator (Power Rule). Hacé Cuentas. https://hacecuentas.com/en/polynomial-derivative-calculator
@misc{hacecuentas_polynomial_derivative_calculator_2026,
author = {Rodríguez, Martín},
title = {{Polynomial Derivative Calculator (Power Rule)}},
year = {2026},
howpublished = {\url{https://hacecuentas.com/en/polynomial-derivative-calculator}},
note = {Hacé Cuentas}
} Content licensed under CC-BY 4.0 — reuse it citing the source with a link to Hacé Cuentas.