https://file.aiquickdraw.com/mathbot/file/1729579586461qau301qf.jpg

From the image you've provided, the problem is asking to find \( \frac{dy}{dx} \) where:

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

You have already begun solving this by introducing \( u = 3x - 2 \), making the expression \( y = u^3 \). Here's how to complete the process step by step:

### Step-by-step solution:

1. **Substitute \( y \) in terms of \( u \):**
   \[
   y = u^3
   \]
   
2. **Differentiate \( y = u^3 \) with respect to \( u \):**
   \[
   \frac{dy}{du} = 3u^2
   \]

3. **Differentiate \( u = 3x - 2 \) with respect to \( x \):**
   \[
   \frac{du}{dx} = 3
   \]

4. **Chain Rule:**
   Using the chain rule, \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \):
   \[
   \frac{dy}{dx} = 3u^2 \times 3 = 9u^2
   \]

5. **Substitute back \( u = 3x - 2 \):**
   \[
   \frac{dy}{dx} = 9(3x - 2)^2
   \]

So, the derivative is:

\[
\frac{dy}{dx} = 9(3x - 2)^2
\]

Would you like more details on any of these steps, or do you have any questions?

### Related questions:

1. How do you apply the chain rule in more complex expressions?
2. Can you explain implicit differentiation and when to use it?
3. How do derivatives of higher-order polynomials differ from simple ones?
4. How do you find the second derivative \( \frac{d^2y}{dx^2} \) for this function?
5. What are some practical applications of this type of derivative?

### Tip:
When using the chain rule, always check for inner functions that can be substituted to simplify the differentiation process!

Learn how to find the derivative of y = (3x - 2)^3 using the chain rule. This step-by-step solution involves substitution, differentiation, and applying the chain rule. Suitable for students in Grades 11-12 studying precalculus or calculus.

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