derivative of `log(sin x^2)`

Answer

To find the derivative of the function \( f(x) = \log(\sin(x^2)) \), we will use the chain rule and the properties of logarithmic differentiation. Here’s the step-by-step solution:

### Step 1: Identify the function and apply the logarithmic derivative
We start with the function:
\[ f(x) = \log(\sin(x^2)) \]
Using the property of logarithmic differentiation, the derivative of \( \log(u) \) is given by:
\[ \frac{d}{dx} \log(u) = \frac{1}{u} \cdot \frac{du}{dx} \]
where \( u = \sin(x^2) \).

### Step 2: Differentiate the outer function
Now we differentiate:
\[ \frac{d}{dx} f(x) = \frac{1}{\sin(x^2)} \cdot \frac{d}{dx}(\sin(x^2)) \]

### Step 3: Differentiate the inner function
Next, we need to find the derivative of \( \sin(x^2) \). We apply 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:
\[ \frac{d}{dx}(x^2) = 2x \]
Thus, we have:
\[ \frac{d}{dx}(\sin(x^2)) = \cos(x^2) \cdot 2x \]

### Step 4: Combine the results
Substituting this back into our expression for \( \frac{d}{dx} f(x) \):
\[ \frac{d}{dx} f(x) = \frac{1}{\sin(x^2)} \cdot (\cos(x^2) \cdot 2x) \]
This simplifies to:
\[ \frac{d}{dx} f(x) = \frac{2x \cos(x^2)}{\sin(x^2)} \]

### Step 5: Use the cotangent identity
We can express this in terms of cotangent:
\[ \frac{d}{dx} f(x) = 2x \cdot \cot(x^2) \]

### Final Result
Thus, the derivative of \( \log(\sin(x^2)) \) is:
\[ \frac{d}{dx} f(x) = 2x \cdot \cot(x^2) \]

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