To find \( f'(2) \), we first need to differentiate the function \( f(x) = \frac{4}{(5 - x)^2} \).

### Step 1: Rewrite the function
We can rewrite \( f(x) \) as:

\[
f(x) = 4(5 - x)^{-2}
\]

### Step 2: Differentiate using the chain rule
Now, apply the chain rule to differentiate \( f(x) \). The chain rule states that if you have a composite function, the derivative is:

\[
\frac{d}{dx} [g(h(x))] = g'(h(x)) \cdot h'(x)
\]

Here, \( g(u) = 4u^{-2} \) and \( h(x) = (5 - x) \). First, differentiate the outer function \( g(u) \) and then multiply by the derivative of the inner function \( h(x) \).

1. Differentiate \( 4(5 - x)^{-2} \):
   \[
   \frac{d}{dx} [4(5 - x)^{-2}] = 4 \cdot (-2)(5 - x)^{-3} \cdot (-1)
   \]

2. Simplify:
   \[
   f'(x) = 8(5 - x)^{-3}
   \]

### Step 3: Substitute \( x = 2 \)
Now, substitute \( x = 2 \) into the derivative:

\[
f'(2) = 8(5 - 2)^{-3} = 8(3)^{-3} = 8 \times \frac{1}{27} = \frac{8}{27}
\]

### Final Answer:
\[
f'(2) = \frac{8}{27}
\]

Would you like further details or clarification?

Here are five related questions:
1. How is the chain rule applied in differentiation?
2. What is the power rule for differentiation?
3. Can we use an alternative method like implicit differentiation here?
4. What is the geometric interpretation of \( f'(x) \)?
5. How do derivatives behave for rational functions?

**Tip:** Always simplify the function before differentiating to make the process easier!

Learn how to differentiate the function f(x) = 4/(5 - x)^2 and find f'(2) using the chain rule. This step-by-step solution is perfect for high school students in Grades 11-12 who are studying calculus and differentiation.

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