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

Transcript text: Use the quotient rule to find the derivative of the function. \[ \begin{array}{l} f(x)=\frac{2 x-5}{5 x+8} \\ f^{\prime}(x)=\square \end{array} \] $\square$

Solution

Solution Steps

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

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

Step 1: Identify the Function

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

Step 2: Compute the Derivatives

We identify the numerator and denominator:

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

Step 3: Apply the Quotient Rule

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: \[ f'(x) = \frac{2(5x + 8) - (2x - 5)(5)}{(5x + 8)^2}. \]

Step 4: Simplify the Expression

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: \[ f'(x) = \frac{41}{(5x + 8)^2}. \]