Differentiate `logsinx^2` with respect to `x` :

Explanation

To differentiate \( \log(\sin(x^2)) \) with respect to \( x \), we will apply the chain rule and the properties of derivatives. Here’s a step-by-step solution:

### Step 1: Identify the outer and inner functions
We have the function \( y = \log(\sin(x^2)) \). Here, the outer function is \( \log(u) \) where \( u = \sin(x^2) \).

### Step 2: Differentiate the outer function
Using the derivative of the logarithmic function:
\[
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