Derivative and Integral Calculator for Polynomials: Tangent Line, Area and Stationary Points

Differentiate and integrate a polynomial and find its turning points.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Enter the coefficients of your polynomial up to degree 5.

Enter the x value for the slope and tangent, and the limits for the area.

The calculator gives the derivatives, the tangent line, the antiderivative, the definite integral, and the maximum and minimum points.

Formulas used

  • \frac{d}{dx}x^n = nx^{n-1}
  • \int x^n\,dx = \frac{x^{n+1}}{n+1} + C
  • y - f(x_0) = f'(x_0)(x - x_0)
  • \int_a^bf\,dx = F(b) - F(a)

Worked example

With the default numbers in the calculator above:

  1. The function: f(x) = x^3 - 6x^2 + 9x + 1 (degree 3)
  2. First derivative: multiply each coefficient by its power and lower the power by 1: f'(x) = 3x^2 - 12x + 9
  3. Second derivative: f''(x) = 6x - 12
  4. Value of f(x) at the x you gave: 3.00
  5. Slope of the curve there (the derivative): -3.00
  6. Tangent line at that point: y = -3 x + 9
  7. Indefinite integral (the antiderivative): raise each power by 1 and divide by the new power: 0.25x^4 - 2x^3 + 4.5x^2 + x + C
  8. Definite integral between the limits (the area under the curve, counted negative below the axis): 12.00
  9. Stationary points (where f'(x) = 0), classified by the sign of the second derivative: x = 1.00, f = 5.00 (local maximum); x = 3.00, f = 1.00 (local minimum)

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: The value of x to evaluate at 4.

  1. The function: f(x) = x^3 - 6x^2 + 9x + 1 (degree 3)
  2. First derivative: multiply each coefficient by its power and lower the power by 1: f'(x) = 3x^2 - 12x + 9
  3. Second derivative: f''(x) = 6x - 12
  4. Value of f(x) at the x you gave: 5.00
  5. Slope of the curve there (the derivative): 9.00
  6. Tangent line at that point: y = 9 x - 31
  7. Indefinite integral (the antiderivative): raise each power by 1 and divide by the new power: 0.25x^4 - 2x^3 + 4.5x^2 + x + C
  8. Definite integral between the limits (the area under the curve, counted negative below the axis): 12.00
  9. Stationary points (where f'(x) = 0), classified by the sign of the second derivative: x = 1.00, f = 5.00 (local maximum); x = 3.00, f = 1.00 (local minimum)

Common questions

How do I differentiate a polynomial?

Multiply each coefficient by its power and reduce the power by one. The derivative of 3x squared is 6x.

How do I find a maximum or minimum?

Set the derivative to zero and solve for x. The second derivative is negative at a maximum and positive at a minimum.

What does the definite integral give?

The area between the curve and the x axis from one limit to the other, with area below the axis counted as negative.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any school or first-year calculus text.

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