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We are asked to find \( f'(4) \) for the function \( f(x) = \ln(4x^2) \).

### Step 1: Differentiate \( f(x) \)
We start by applying the chain rule to differentiate \( f(x) = \ln(4x^2) \).

The derivative of \( \ln(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \), where \( u = 4x^2 \).

First, differentiate \( 4x^2 \):
\[
\frac{d}{dx}(4x^2) = 8x
\]

Now, apply the chain rule:
\[
f'(x) = \frac{1}{4x^2} \cdot 8x = \frac{8x}{4x^2} = \frac{2}{x}
\]

### Step 2: Evaluate at \( x = 4 \)
Now, substitute \( x = 4 \) into the derivative \( f'(x) = \frac{2}{x} \):
\[
f'(4) = \frac{2}{4} = \frac{1}{2}
\]

### Final Answer:
\[
f'(4) = \frac{1}{2}
\]

Would you like further details or have any questions?

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Here are 5 related questions to explore:
1. How do you apply the chain rule in other logarithmic functions?
2. What happens if the base of the logarithm changes from natural log to common log?
3. How would you differentiate \( \ln(ax^b) \) for different values of \(a\) and \(b\)?
4. Can you generalize this result to other power functions inside logarithms?
5. How would you approach finding the second derivative \( f''(x) \)?

**Tip:** Always simplify the function inside a logarithm before differentiating, as it often makes the process easier.

Find f'(4) for f(x) = ln(4x^2). Answer as an exact fraction or round to at least 2 decimal places.

Learn how to find the derivative and evaluate f'(4) for the function f(x) = ln(4x^2). The solution covers key calculus concepts like the chain rule and differentiation of logarithmic functions. Suitable for students in Grades 11-12.

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