Questions: Use the quotient rule to find the derivative of the function. f(x)=(2x-5)/(5x+8) f'(x)=

Solution Steps

To find the derivative of the function $f(x) = \frac{2x - 5}{5x + 8}$ using the quotient rule, we follow these steps:

1. Identify the numerator $u(x) = 2x - 5$ and the denominator $v(x) = 5x + 8$.
2. Compute the derivatives $u'(x)$ and $v'(x)$.
3. Apply the quotient rule: $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$.

We start with the function given by $$f(x) = \frac{2x - 5}{5x + 8}.$$

We identify the numerator and denominator:

• $u(x) = 2x - 5$ with derivative $u'(x) = 2$,
• $v(x) = 5x + 8$ with derivative $v'(x) = 5$.

Using the quotient rule, we find the derivative $f'(x)$: $$f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}.$$ Substituting the values we computed: $$f'(x) = \frac{2(5x + 8) - (2x - 5)(5)}{(5x + 8)^2}.$$

Calculating the numerator: $$2(5x + 8) = 10x + 16,$$ $$(2x - 5)(5) = 10x - 25.$$ Thus, the numerator becomes: $$10x + 16 - (10x - 25) = 10x + 16 - 10x + 25 = 41.$$ Therefore, we have: $$f'(x) = \frac{41}{(5x + 8)^2}.$$

Final Answer