Question

(c) Are the expressions for the derivative in (a) and (b) the same?

(a) Find the derivative of f(x) = (x²+5)(2x-5) by first expanding the polynomials. Enter the fully simplified expression for f(x) after expanding the polynomials. f(x) =

Answer 1

The derivative of the function f(x) = (x²+5)(2x-5) when expanded is f'(x) = 6x² - 10x + 10, after applying the power rule for differentiation to each term.

Step-by-step explanation:

To find the derivative of the function f(x) = (x²+5)(2x-5), we must first expand the polynomial. By distributing the terms, we get:

f(x) = x²(2x) + x²(-5) + 5(2x) - 5(5)

f(x) = 2x³ - 5x² + 10x - 25

Now, we take the derivative of each term separately, applying the basic power rule of differentiation:

f'(x) = 3(2x²) - 2(5x) + 10

f'(x) = 6x² - 10x + 10

This is the fully simplified expression for the derivative of f(x). It should be noted that whether we expand the polynomial first or use the product rule, the expressions for the derivative in parts (a) and (b) will ultimately be the same, as differentiation is a linear operation.