Question

use the quotient rule to find the derivative of the function. f(x) = (4x - 5)/(8x + 7) f’(x) = \boxed{}

Explanation:

Step1: Recall Quotient Rule

The quotient rule states that for a function \( f(x) = \frac{g(x)}{h(x)} \), the derivative \( f'(x) \) is given by \( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2} \), where \( g(x) = 4x - 5 \) and \( h(x) = 8x + 7 \).

Step2: Find \( g'(x) \) and \( h'(x) \)

Differentiate \( g(x) = 4x - 5 \) with respect to \( x \): \( g'(x) = 4 \) (since the derivative of \( 4x \) is 4 and the derivative of -5 is 0).
Differentiate \( h(x) = 8x + 7 \) with respect to \( x \): \( h'(x) = 8 \) (since the derivative of \( 8x \) is 8 and the derivative of 7 is 0).

Step3: Apply Quotient Rule

Substitute \( g(x) \), \( g'(x) \), \( h(x) \), and \( h'(x) \) into the quotient rule formula:

$$
LATEXBLOCK0
$$

Answer:

\(\frac{68}{(8x + 7)^2}\)