Polynomial Derivative & Integral Calculator

Understanding the polynomial derivative and integral calculator — concept and principle

This calculator differentiates and integrates a polynomial such as 3x³ − 2x² + 5x + 1. You enter the coefficients from the highest power down, and it returns the first derivative, the second derivative and the antiderivative (indefinite integral). Polynomial derivatives give slopes, velocities and rates of change; second derivatives describe acceleration and curvature; antiderivatives give areas, distances and accumulated totals.

How it is calculated

For a polynomial p(x) = cₙxⁿ + … + c₁x + c₀, each term is handled separately:

• Derivative (power rule): d/dx (cxⁿ) = n·c·xⁿ⁻¹. The constant term disappears.
• Second derivative: apply the rule twice: n(n − 1)·c·xⁿ⁻².
• Antiderivative: ∫cxⁿ dx = c·xⁿ⁺¹ ÷ (n + 1), plus an arbitrary constant C.

Differentiation and integration are linear, so you can work term by term and add the results.

The principle

The derivative is the limit of the slope of a secant line, [p(x + h) − p(x)] ÷ h, as h → 0. For xⁿ, expanding (x + h)ⁿ with the binomial theorem gives xⁿ + n·xⁿ⁻¹h + (terms with h² and higher). Subtracting xⁿ, dividing by h and letting h shrink leaves exactly n·xⁿ⁻¹. Integration reverses this: differentiating xⁿ⁺¹ ÷ (n + 1) gives back xⁿ. Because any constant has derivative 0, every antiderivative is only determined up to + C; this is the content of the fundamental theorem of calculus.

Worked example

1. Default input: 3 −2 5 1, so p(x) = 3x³ − 2x² + 5x + 1.
2. Derivative: 3·3x² − 2·2x + 5 = 9x^2 − 4x + 5 (the constant 1 drops out).
3. Second derivative: differentiate again: 9·2x − 4 = 18x − 4.
4. Antiderivative: 3x⁴/4 − 2x³/3 + 5x²/2 + 1x = 0.75x^4 − 0.6666666667x^3 + 2.5x^2 + 1x + C.

How to read the result

p′(x) is the slope of the graph at x: positive means rising, negative means falling, zero marks a possible maximum or minimum. Here p′(x) = 9x² − 4x + 5 has discriminant 16 − 180 < 0, so it is always positive and p is increasing everywhere. p″(x) = 18x − 4 changes sign at x = 2/9, the inflection point where the curve switches from bending down to bending up. To get a definite integral, evaluate the antiderivative at the two limits and subtract.

Common mistakes

• Forgetting the constant of integration C.
• Leaving out zero coefficients: x³ + 1 must be entered as 1 0 0 1.
• Dividing by n instead of n + 1 when integrating.
• Reading 0.6666666667 as an exact value; it is the rounded decimal of 2/3.

Frequently asked questions

What does the second derivative tell me?

It measures how the slope changes. p″ > 0 means the graph is concave up, p″ < 0 concave down.

How do I find the value of C?

Use a known point. If you know F(0) = 2, then C = 2 because every other term vanishes at x = 0.

What is the derivative of a constant?

Zero, because a constant does not change; its graph is a horizontal line.

Sources: Stewart, J., Calculus: Early Transcendentals, §3.1 Derivatives of Polynomials and Exponential Functions · Apostol, T. M., Calculus, Vol. 1, Wiley