Differentiating polynomials is the first thing most people do in calculus, and it rests on a single rule. This calculator applies the power rule term by term, shows each term’s conversion, and then puts the derivative to work: it evaluates the function and its slope at a point, writes the tangent line, and finds critical points when the derivative is linear or quadratic.
How to use the polynomial derivative calculator
- Type the polynomial, for example
3x^4 - 2x^3 + 5x - 7. Expressions such as(x + 1)^3are expanded first. - Choose which derivative you want, from the first up to the fifth.
- Enter the x value where you want the value, slope and tangent line.
- Read the derivative on the tape. The steps show the power rule applied to every term, and the graph draws the curve with its tangent line.
All arithmetic is exact, so fractional coefficients such as ⅓x3 stay as fractions.
Derivative rules for polynomials
Those three facts are enough for any polynomial. The tangent line at x0 is
Worked example
Let f(x) = 3x4 − 2x3 + 5x − 7.
Term by term: 3x4 → 12x3; −2x3 → −6x2; 5x → 5; −7 → 0.
Derivative: f′(x) = 12x3 − 6x2 + 5.
At x = 1: f(1) = 3 − 2 + 5 − 7 = −1 and f′(1) = 12 − 6 + 5 = 11.
Tangent line: y − (−1) = 11(x − 1), so y = 11x − 12.
Differentiating again gives f″(x) = 36x2 − 12x, which measures how quickly the slope itself changes.
Using derivatives to find maxima and minima
At a smooth peak or valley, the tangent line is horizontal, so f′(x) = 0. For f(x) = x3 − 3x, the derivative 3x2 − 3 is zero at x = ±1. The second derivative 6x is positive at x = 1 (a local minimum, f = −2) and negative at x = −1 (a local maximum, f = 2). The calculator performs this test automatically whenever f′ is linear or quadratic, giving exact roots such as ±√6/3 when they are irrational.
Not every critical point is an extreme value. For x3 + 3x2 + 3x = (x + 1)3 − 1, the slope touches zero at x = −1 but the function keeps increasing on both sides.
What the derivative means in practice
| If f measures… | then f′ measures… |
|---|---|
| position over time | velocity |
| velocity over time | acceleration |
| total cost for q units | marginal cost of one more unit |
| area of a square of side s | how fast the area grows as s grows (2s) |
Common mistakes
- Dropping the coefficient. The derivative of 5x3 is 15x2, not 3x2.
- Keeping the constant. Constant terms vanish.
- Confusing value and slope. f(x0) is the height of the curve; f′(x0) is its steepness.
For the reverse process — area under a curve — use the definite integral calculator. To see how values change across an interval, build a function table, and for the slope of a straight line through two points use the slope calculator.
Frequently asked questions
What is the power rule?
The derivative of a·xⁿ is n·a·xⁿ⁻¹: multiply by the exponent, then lower the exponent by one. Constants have derivative 0, and the derivative of a sum is the sum of the derivatives.
What does the derivative at a point tell me?
It is the slope of the tangent line there, or the instantaneous rate of change. If f(t) is position, f′(t) is velocity at time t.
How is the tangent line found?
Use the point (x₀, f(x₀)) and slope m = f′(x₀) in point-slope form, y − f(x₀) = m(x − x₀), then simplify to y = mx + b.
What are critical points?
Values of x where f′(x) = 0. They are candidates for local maxima and minima. A positive second derivative there means a local minimum and a negative one means a local maximum.
Why does a high-order derivative become zero?
Each derivative lowers the degree by one, so a polynomial of degree n has a constant n-th derivative and every derivative after that is 0.