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We need to find the derivative of the given function \( f(x) = \frac{5x + 2}{5x - 2} \) and then evaluate \( f'(5) \).

### Step 1: Differentiate the function
This is a rational function, so we will use the quotient rule for differentiation. The quotient rule states:

\[
f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}
\]

where \( g(x) \) is the numerator, and \( h(x) \) is the denominator.

- \( g(x) = 5x + 2 \) 
- \( h(x) = 5x - 2 \)

Now, differentiate both \( g(x) \) and \( h(x) \):

- \( g'(x) = 5 \)
- \( h'(x) = 5 \)

### Step 2: Apply the quotient rule
Substitute into the quotient rule:

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

Simplify the numerator:

\[
f'(x) = \frac{25x - 10 - 25x - 10}{(5x - 2)^2}
\]

\[
f'(x) = \frac{-20}{(5x - 2)^2}
\]

### Step 3: Evaluate \( f'(5) \)
Now substitute \( x = 5 \) into the derivative:

\[
f'(5) = \frac{-20}{(5(5) - 2)^2} = \frac{-20}{(25 - 2)^2} = \frac{-20}{23^2} = \frac{-20}{529}
\]

Thus, \( f'(5) = \frac{-20}{529} \).

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Would you like more details or clarifications on any part? 

Here are some related questions:
1. What is the quotient rule for differentiation?
2. How do you simplify the result from applying the quotient rule?
3. Can the chain rule be applied to rational functions?
4. How do we interpret the result \( f'(x) \) in terms of the slope of the tangent line?
5. How do the numerator and denominator impact the overall behavior of a rational function?

**Tip**: Always check for ways to simplify the function before applying complex differentiation rules.

Find the derivative of the function f(x) = (5x + 2)/(5x - 2) and evaluate f'(x) at x = 5.

Learn how to apply the quotient rule to differentiate f(x) = (5x + 2)/(5x - 2) and evaluate the derivative at x = 5. This problem covers key concepts in calculus, including rational function differentiation.

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