`log(sin x^2)`

Answer

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

### Step 1: Identify the function
Let \( y = \log(\sin(x^2)) \).

### Step 2: Apply the chain rule
Using the chain rule for differentiation, we have:
\[ \frac{dy}{dx} = \frac{1}{\sin(x^2)} \cdot \frac{d}{dx}(\sin(x^2)) \]

### Step 3: Differentiate \( \sin(x^2) \)
Now we need to differentiate \( \sin(x^2) \). Using the chain rule again:
\[ \frac{d}{dx}(\sin(x^2)) = \cos(x^2) \cdot \frac{d}{dx}(x^2) \]
The derivative of \( x^2 \) is \( 2x \). Therefore:
\[ \frac{d}{dx}(\sin(x^2)) = \cos(x^2) \cdot 2x \]

### Step 4: Substitute back into the derivative
Now substitute this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{\sin(x^2)} \cdot (\cos(x^2) \cdot 2x) \]

### Step 5: Simplify the expression
This simplifies to:
\[ \frac{dy}{dx} = \frac{2x \cos(x^2)}{\sin(x^2)} \]

### Step 6: Use the cotangent identity
Recall that \( \cot(x) = \frac{\cos(x)}{\sin(x)} \). Thus:
\[ \frac{dy}{dx} = 2x \cot(x^2) \]

### Final Answer
The derivative of \( y = \log(\sin(x^2)) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 2x \cot(x^2) \]

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