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We are given the function \( f(x) = 4(8 - 6x)^4 \), and the task is to find the second derivative, \( f''(x) \).

### Step 1: First Derivative
To find \( f'(x) \), we use the chain rule. 

Let \( u = 8 - 6x \), so that \( f(x) = 4u^4 \).

1. Differentiate the outer function \( 4u^4 \):
   \[
   \frac{d}{du}[4u^4] = 16u^3
   \]

2. Now, differentiate the inner function \( u = 8 - 6x \):
   \[
   \frac{d}{dx}[8 - 6x] = -6
   \]

Using the chain rule:
\[
f'(x) = 16u^3 \cdot (-6) = -96(8 - 6x)^3
\]

### Step 2: Second Derivative
Now, to find the second derivative \( f''(x) \), we again use the chain rule to differentiate \( f'(x) = -96(8 - 6x)^3 \).

1. Differentiate the outer function \( -96(8 - 6x)^3 \):
   \[
   \frac{d}{du}[-96u^3] = -288u^2
   \]

2. Again, differentiate \( u = 8 - 6x \):
   \[
   \frac{d}{dx}[8 - 6x] = -6
   \]

Using the chain rule:
\[
f''(x) = -288(8 - 6x)^2 \cdot (-6) = 1728(8 - 6x)^2
\]

Thus, the second derivative is:
\[
f''(x) = 1728(8 - 6x)^2
\]

Would you like a detailed explanation of any steps or more examples?

### Follow-up questions:
1. How is the chain rule applied in multivariable functions?
2. What is the general formula for finding higher-order derivatives?
3. How does the second derivative relate to concavity of a function?
4. What are other methods for checking the accuracy of derivatives?
5. Can you provide an example of a second derivative test for critical points?

**Tip:** The chain rule is useful whenever a function is composed of multiple layers—always break it down step by step.

Find the second derivative of the function: f(x) = 4(8 - 6x)^4

Find the second derivative of f(x) = 4(8 - 6x)^4 using the chain rule. This problem involves calculating first and second derivatives step by step, suitable for high school and college-level calculus students.

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