1.1: 5.6 Derivatives of Polynomials

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Definition: Derivative of Polynomials

Given the function f(x) = 6x²+ 3x + 6, find the derivative. When the function is a polynomial, you can find the derivative by doing the following for each term.

f'(x) = 2(6)x^2-1+ (1)(3)x⁰ + (0)6x⁰^-1 = 12x¹ + 3x⁰+ 0 = 12x + 3

1) Bring the exponent down and multiply it by the coefficient.

2) Subtract one from the exponent.

3) Simplify.

4) The derivative of a constant is 0.

Example \(\PageIndex{5.6.1}\)

Find the derivative of the following polynomial.

f(x) = 4x³- 5x² + 10x - 5

Solution

f'(x) = 3(4)x^{3 - 1} - (2)(5)x^{2 - 1} + (1)(10)x^{1 - 1} - 0

f'(x) = 12x² - 10x¹+ 10x⁰ - 0

f'(x) = 12x²- 10x + 10

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