Polynomial Calculator
Give this calculator the coefficients of any single-variable polynomial and a value of x, and it evaluates P(x), builds the derivative f′(x) term by term with the power rule, evaluates f′(x) at your point, and computes the exact definite integral from your x to an upper bound you choose. One entry, four answers, with every differentiation step printed so you can check the work.
Single-variable polynomials with real coefficients only. The integral uses the exact antiderivative, not a numerical approximation.
Quick Answer
For coefficients 3, 2, -5, 7 (that is, P(x) = 3x³ + 2x² – 5x + 7) at x = 2: P(2) = 29.0000, the derivative is f′(x) = 9x² + 4x – 5, so f′(2) = 39.0000. Add an upper bound of b = 2 with x = 0 and the exact definite integral from 0 to 2 is 21.3333. The calculator above reproduces all four numbers.
Key takeaways
- Differentiation works term by term: d/dx [ax^n] = anx^(n-1). Constants vanish because a constant never changes.
- f′(x) at a point is the slope of the curve there. f′(2) = 39 means the polynomial is climbing steeply through x = 2.
- The integral is computed exactly from the antiderivative, not approximated numerically: each term ax^n integrates to ax^(n+1)/(n+1).
- Coefficients go in highest degree first, comma separated: "3, 2, -5, 7" means 3x^3 + 2x^2 – 5x + 7. No expression syntax to learn.
- Quick Answer
- How to use the polynomial calculator
- The power rule, term by term
- Worked example: 3x³ + 2x² – 5x + 7 at x = 2
- What one entry produces
- What the derivative value tells you
- The integral is exact, not approximated
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
How to use the polynomial calculator
Enter three things and click Calculate. The results stay hidden until you do.
- Enter the coefficients, highest degree first: "3, 2, -5, 7" is pre-filled as the worked example. Any real numbers work, including negatives and decimals such as "1.5, -2, 0.25". Spaces around the commas are fine; anything that is not a number gets a specific error naming the bad entry.
- Enter the value of x: the point where P(x) and f′(x) are evaluated. This is also the lower bound of the definite integral.
- Optionally enter the upper bound b: leave it empty to skip the integral, or fill it to get the exact definite integral from x to b.
The results show the polynomial as entered, its degree, P(x) and f′(x) evaluated at your point, the power-rule steps behind the derivative, and the integral when you gave a bound. Reset restores the worked-example values.
The power rule, term by term
Differentiation treats each term separately. For any term ax^n, multiply the coefficient by the power and reduce the power by one; do that for every term and add the results. That is the whole of the power rule, as taught in the Khan Academy differentiation unit and in OpenStax Calculus, section 3.3:
3x³ → 9x² | 2x² → 4x | –5x → –5 | 7 → 0
A cubic term becomes quadratic, a linear term becomes a constant, and constants vanish: the derivative of 7 is 0 because 7 never changes as x changes. The tool prints every one of these steps, so you can audit it against your own working instead of trusting a single answer line.
Worked example: 3x³ + 2x² – 5x + 7 at x = 2
The coefficients 3, 2, -5, 7 and x = 2 are pre-filled. Click Calculate and check each line:
| Term | Power-rule step | Contribution |
|---|---|---|
| 3x³ | 3 · 3 · x^(3–1) | 9x² |
| 2x² | 2 · 2 · x^(2–1) | 4x |
| –5x | –5 · 1 · x^(1–1) | –5 |
| 7 | constant | 0 |
For the integral, set x = 0 and upper bound b = 2. The antiderivative is 3x⁴/4 + 2x³/3 – 5x²/2 + 7x, evaluated from 0 to 2: 12 + 5.3333 – 10 + 14 = 21.3333. Every row of the table above matches what the tool prints, so type any of these in and you get the same figures back.
What one entry produces
A single coefficient list feeds three independent computations. Nothing is shared between them except the input:
What the derivative value tells you
f′(x) at a point is the slope of the curve there: the instantaneous rate of change. The derivative is formally defined as a limit, which a limit calculator can explore; the power rule is simply the shortcut that limit works out to for polynomials.
Concretely, f′(2) = 39 means the polynomial is climbing steeply through x = 2, gaining about 39 units of y per unit of x. Our average slope calculator covers the secant-line version of the same idea; the derivative is what that average slope becomes as the interval shrinks to a point. Where f′(x) = 0 the curve flattens: those stationary points are where maxima and minima live, which is why the sign of the derivative is the standard test for increasing and decreasing intervals.
The integral is exact, not approximated
Numerical-integration tools sample the curve and add up slices. This tool integrates the antiderivative exactly: each term ax^n becomes ax^(n+1)/(n+1), evaluated at b and x and subtracted, following the Fundamental Theorem of Calculus (OpenStax Calculus, section 5.3). The lower bound is always your x, so the integral answers "accumulated change from x to b". If you work with antiderivatives on their own, our general antiderivative calculator covers the indefinite case.
What competing calculators miss
Polynomial tools split into four camps, and each drops something this page keeps:
- Expression-based differentiators (Softonic's polynomial derivative tool): type 3x^3 and get f′(x) with steps. Good teaching, but no evaluation at a point and no integral: you get the formula, not the numbers.
- Term-box calculators (calculator.city's Materia Calculo 1): coefficient and exponent boxes per term, and they even accept negative and fractional exponents, which this page honestly does not. But they differentiate only: no P(x), no f′(x) value, no integral.
- General symbolic tools (bravecalculator, derivativecalculus.com): full CAS engines handling trig, products and higher orders. Powerful, and overkill for a polynomial: you trade a simple coefficient list for expression syntax and option panels.
- WolframAlpha: does everything, in natural language, if you learn its input conventions and tolerate the ad-supported free tier.
- The gap this page fills: comma-separated coefficients in, four answers out: P(x), the term-by-term derivative, f′(x) at your point, and the exact definite integral. No syntax to learn, no separate tools for evaluation and differentiation, no numerical approximation where an exact answer exists.
One honest boundary: this is a single-variable polynomial tool with real coefficients. Negative and fractional exponents, multiple variables and non-polynomial functions are outside its scope; the step-by-step symbolic tools above cover those.
Frequently asked questions
As comma-separated coefficients, highest degree first: "3, 2, -5, 7" means 3x^3 + 2x^2 – 5x + 7. There is no expression syntax to learn: just the numbers in order.
Multiply by the power, then reduce the power by one: d/dx [ax^n] = anx^(n–1). So 3x^3 differentiates to 9x^2, and a linear term –5x becomes the constant –5.
It is the slope of the curve at that point: the instantaneous rate of change. f′(2) = 39 means the polynomial is climbing steeply through x = 2, gaining about 39 units of y per unit of x.
Exactly, via the antiderivative: each term ax^n integrates to ax^(n+1)/(n+1), evaluated at the upper bound b and the lower bound x and subtracted. No numerical approximation is involved.
You get P(x), the derivative f′(x), and f′(x) evaluated at your point, without the integral. The integral appears only when both x and b are filled in.
Yes. Any real numbers work, for example "1.5, -2, 0.25", which at x = 2 gives P(2) = 2.2500 and f′(2) = 4.0000. Negative and fractional exponents are not supported, only coefficients.
A constant never changes as x changes, so its rate of change is zero. That is why the "+ 7" disappears from f′(x): d/dx 7 = 0.
The highest power with a nonzero coefficient, which the tool reports alongside the result. With n coefficients entered highest degree first, the degree is n – 1.
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References
- Khan Academy. Power rule (differentiation). Accessed October 23, 2026.
- OpenStax. Calculus Volume 1, 3.3 Differentiation Rules. Accessed October 23, 2026.
- OpenStax. Calculus Volume 1, 5.3 The Fundamental Theorem of Calculus. Accessed October 23, 2026.
- Weisstein, E. W. Polynomial, Wolfram MathWorld. Accessed October 23, 2026.
- Massachusetts Institute of Technology. 18.01 Single Variable Calculus, MIT OpenCourseWare. Accessed October 23, 2026.