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The problem presented is:

\[
y = (3x - 2)^4
\]

You are asked to find the second derivative \(\frac{d^2y}{dx^2}\).

### Step 1: First Derivative \(\frac{dy}{dx}\)

We will first apply the chain rule to find the first derivative.

\[
\frac{dy}{dx} = \frac{d}{dx} \left( (3x - 2)^4 \right)
\]

Using the chain rule:

\[
\frac{dy}{dx} = 4(3x - 2)^3 \cdot \frac{d}{dx}(3x - 2)
\]

The derivative of \(3x - 2\) is just 3, so:

\[
\frac{dy}{dx} = 4(3x - 2)^3 \cdot 3
\]

Simplifying:

\[
\frac{dy}{dx} = 12(3x - 2)^3
\]

### Step 2: Second Derivative \(\frac{d^2y}{dx^2}\)

Next, we differentiate \(\frac{dy}{dx} = 12(3x - 2)^3\) again with respect to \(x\), applying the chain rule once more.

\[
\frac{d^2y}{dx^2} = \frac{d}{dx} \left( 12(3x - 2)^3 \right)
\]

Applying the chain rule:

\[
\frac{d^2y}{dx^2} = 12 \cdot 3(3x - 2)^2 \cdot \frac{d}{dx}(3x - 2)
\]

Again, the derivative of \(3x - 2\) is 3, so:

\[
\frac{d^2y}{dx^2} = 12 \cdot 3(3x - 2)^2 \cdot 3
\]

Simplifying:

\[
\frac{d^2y}{dx^2} = 108(3x - 2)^2
\]

### Final Answer:

\[
\frac{d^2y}{dx^2} = 108(3x - 2)^2
\]

Would you like more details or have any questions? Here are 5 related questions to expand your understanding:

1. How do you apply the chain rule in general for composite functions?
2. What is the significance of higher-order derivatives in physics and engineering?
3. How do you compute the third derivative of this function?
4. What happens when you differentiate a power function like \(x^n\)?
5. What are some common mistakes to avoid when applying the chain rule?

**Tip:** When differentiating powers of functions, always remember to apply the chain rule carefully, and ensure that you differentiate the inner function correctly.

Learn how to find the second derivative of y = (3x - 2)^4 step-by-step using the chain rule. This calculus problem demonstrates the process of differentiation and applying the chain rule. Suitable for advanced high school or college students.

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