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.