Polynomial Derivative Calculator (Power Rule)
Enter your polynomial coefficients from highest to lowest degree (comma-separated) and this calculator applies the Power Rule to produce the exact derivative expression instantly. For 3x²+2x+1 enter 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.
The Power Rule for polynomials: d/dx [axⁿ] = n·axⁿ⁻¹. For each term, multiply the coefficient by its exponent and reduce the exponent by 1; the constant term becomes 0. Example: derivative of 3x²+2x+1 is **6x+2**; derivative of x³−3x+5 is **3x²−3**.
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.
Worked Example
- Polynomial: 3x²+2x+1 → enter coefficients
3, 2, 1 - Term 3x²: exponent 2 → 2×3 = 6, new exponent 1 → 6x
- Term 2x: exponent 1 → 1×2 = 2, new exponent 0 → +2
- Term 1 (constant): exponent 0 → 0×1 = 0 → vanishes
- Result: p′(x) = 6x+2
How it works
2 min readHow 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.
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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 |
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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.
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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.
Frequently asked questions
What is the Power Rule for polynomial derivatives?
The Power Rule states d/dx[axⁿ] = n·axⁿ⁻¹: multiply the coefficient by the exponent and reduce the exponent by 1. For a polynomial you apply it term by term using the Sum Rule. Example: d/dx[3x²+2x+1] = 6x+2. It is derived from the limit definition of the derivative and holds for all real-number exponents.
How do I enter coefficients for a polynomial with missing terms?
You must include a 0 coefficient for every missing power. For 4x³ − 9x (missing x² and the constant), enter 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?
The constant term a₀ (coefficient of x⁰) always becomes 0. Because its exponent is 0, the Power Rule gives 0·a₀·x⁻¹ = 0. Geometrically, a constant shifts the graph up or down but has no effect on slope — confirming its derivative is zero.
What is the derivative of a constant polynomial like f(x) = 7?
The derivative is 0. A constant function has a horizontal graph with slope 0 everywhere. Enter 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?
Run the calculator iteratively. Enter your polynomial to get p′(x), then re-enter the result's coefficients to get p″(x), and so on. Example: p(x)=x⁴ → coefficients 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?
If position is s(t), then velocity v(t) = s′(t) and acceleration a(t) = v′(t) = s″(t). For s(t)=−16t²+64t+80 (free-fall in feet): v(t)=−32t+64 (zero at t=2 s, the peak) and a(t)=−32 ft/s² (gravitational acceleration near Earth's surface).
Does the Power Rule apply to negative or fractional exponents?
Yes — d/dx[xⁿ]=n·xⁿ⁻¹ holds for all real n, including negatives and fractions (e.g., d/dx[x^(1/2)]=(1/2)x^(−1/2)). However, this calculator targets standard polynomials with non-negative integer exponents; fractional-exponent terms fall outside polynomial scope.
Why does differentiating always reduce the polynomial's degree by exactly 1?
Every term axⁿ becomes n·axⁿ⁻¹ — the exponent drops by 1. The leading term aₙxⁿ becomes n·aₙxⁿ⁻¹, so the highest power in p′(x) is n−1. The only exception is a constant polynomial (degree 0), whose derivative is the zero polynomial (degree undefined or −∞). This is why n+1 repeated differentiations of a degree-n polynomial reduce it to zero.