Polynomial Derivative and Integral Calculator

How it works

This one handles polynomials, using the power rule on each term: xⁿ becomes nxⁿ⁻¹ when you differentiate, and xⁿ⁺¹/(n + 1) when you integrate. Type powers with ^, like 3x^3 − 2x + 5. It doesn’t do trig, logs or the chain rule.

Example

Differentiate 4x³ − 6x² + 2x − 7, and find the slope at x = 2.

f′(x) = 12x² − 12x + 2

Integral

x⁴ − 2x³ + x² − 7x + C

f(2)

5

f′(2)

26

Steps

1. 1
   4x³: bring down 3 and lower the power: 12x².
2. 2
   −6x²: bring down 2 and lower the power: −12x.
3. 3
   2x: bring down 1 and lower the power: 2.
4. 4
   −7 is a constant, so its derivative is 0.
5. 5
   At x = 2: slope f′(2) = 26.
6. 6
   For the integral, raise each power by 1 and divide by the new power: x⁴ − 2x³ + x² − 7x + C.

More examples

Find the slope of x² − 4x at x = 2.

f′(x) = 2x − 4

Integral

1/3x³ − 2x² + C

f(2)

−4

f′(2)

0

Steps

1. 1
   x²: bring down 2 and lower the power: 2x.
2. 2
   −4x: bring down 1 and lower the power: −4.
3. 3
   At x = 2: slope f′(2) = 0.
4. 4
   For the integral, raise each power by 1 and divide by the new power: 1/3x³ − 2x² + C.

Differentiate and integrate x⁵ − 3x + 8, slope at x = 1.

f′(x) = 5x⁴ − 3

Integral

1/6x⁶ − 3/2x² + 8x + C

f(1)

6

f′(1)

2

Steps

1. 1
   x⁵: bring down 5 and lower the power: 5x⁴.
2. 2
   −3x: bring down 1 and lower the power: −3.
3. 3
   8 is a constant, so its derivative is 0.
4. 4
   At x = 1: slope f′(1) = 2.
5. 5
   For the integral, raise each power by 1 and divide by the new power: 1/6x⁶ − 3/2x² + 8x + C.

Common mistake

The derivative of a constant is 0, and an indefinite integral always needs + C.

Questions

What does the slope at a point mean?

It is how fast the function is changing there, and the slope of the tangent line. A slope of 0 is a flat spot, often a peak or a valley.

How do I find where the slope is zero?

Set the derivative equal to 0 and solve. For x² − 4x, 2x − 4 = 0 gives x = 2.

Why does the integral have + C?

Every constant has a derivative of 0, so x² + 5 and x² − 1 have the same derivative. C stands for whichever constant it was.

Keep going