Let's solve the derivative step by step. 

You are given the function:

\[
\frac{d}{dx}(y) = \frac{d}{dx} \left[\left(\sin(\ln(x^2))\right)^2\right]
\]

We'll use the chain rule to differentiate this expression.

### Step 1: Apply the Chain Rule
Let \( u = \sin(\ln(x^2)) \), so the expression becomes \( u^2 \).

Now, differentiate with respect to \( x \):

\[
\frac{d}{dx} \left( u^2 \right) = 2u \cdot \frac{du}{dx}
\]

### Step 2: Differentiate \( u \) with respect to \( x \)
Now, differentiate \( u = \sin(\ln(x^2)) \) with respect to \( x \):

\[
\frac{du}{dx} = \cos(\ln(x^2)) \cdot \frac{d}{dx}(\ln(x^2))
\]

### Step 3: Differentiate \( \ln(x^2) \) with respect to \( x \)
The derivative of \( \ln(x^2) \) is:

\[
\frac{d}{dx}(\ln(x^2)) = \frac{1}{x^2} \cdot \frac{d}{dx}(x^2) = \frac{1}{x^2} \cdot 2x = \frac{2}{x}
\]

So,

\[
\frac{du}{dx} = \cos(\ln(x^2)) \cdot \frac{2}{x}
\]

### Step 4: Substitute Back
Now substitute \( \frac{du}{dx} \) back into the expression from Step 1:

\[
\frac{d}{dx} \left[\left(\sin(\ln(x^2))\right)^2\right] = 2\sin(\ln(x^2)) \cdot \cos(\ln(x^2)) \cdot \frac{2}{x}
\]

### Step 5: Simplify the Expression
We can use the trigonometric identity \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \) to simplify the expression:

\[
\frac{d}{dx} \left[\left(\sin(\ln(x^2))\right)^2\right] = \frac{2 \cdot 2}{x} \cdot \sin(\ln(x^2))\cos(\ln(x^2)) = \frac{2 \sin(2\ln(x^2))}{x}
\]

Thus, the final derivative is:

\[
\frac{d}{dx}(y) = \frac{2 \sin(2\ln(x^2))}{x}
\]

### Summary:
The derivative of \( y = \left(\sin(\ln(x^2))\right)^2 \) with respect to \( x \) is:

\[
\frac{d}{dx}(y) = \frac{2 \sin(2\ln(x^2))}{x}
\]

Would you like to explore any part of this solution further, or have any specific questions?

Here are 5 related questions you might find interesting:

1. How would the derivative change if the function inside the sine was \( \ln(x) \) instead of \( \ln(x^2) \)?
2. What is the second derivative of \( y = \left(\sin(\ln(x^2))\right)^2 \)?
3. How does the chain rule apply when there are three nested functions instead of two?
4. Can you integrate the function \( y = \left(\sin(\ln(x^2))\right)^2 \)?
5. What happens if the argument inside the sine function is \( \ln(x^n) \) where \( n \) is any constant?

**Tip:** Understanding the chain rule is crucial for differentiating composite functions. Breaking down the problem into smaller steps often makes complex derivatives more manageable.

Learn how to differentiate y = (sin(ln(x^2)))^2 using the chain rule and trigonometric identities. This problem explores differentiation techniques involving composite functions and trigonometric functions. Suitable for undergraduate level students.

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