If `y = (sin x)^(log x) " then " (dy)/(dx) =`?

Options

A. `(log x).(sinx)^((log x -1)).cos x`
B. `(sin x)^(log x).{(x log x + log sin x)/(x)}`
C. `(sin x)^(log x).{((x log x) cot x + log sin x)/(x)}`
D. none of these

Explanation

To find the derivative of the function \( y = (\sin x)^{\log x} \), we can follow these steps:

### Step 1: Take the logarithm of both sides
We start by taking the natural logarithm of both sides:
\[ \log y = \log((\sin x)^{\log x}) \]

### Step 2: Use the property of logarithms
Using the property of logarithms that states \( \log(a^b) = b \log a \), we can rewrite the equation:
\[ \log y = \log x \cdot \log(\sin x) \]

### Step 3: Differentiate both sides
Now, we differentiate both sides with respect to \( x \). Using implicit differentiation on the left side and the product rule on the right side:
\[ \frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(\log x \cdot \log(\sin x)) \]

### Step 4: Apply the product rule
Using the product rule on the right side:
\[ \frac{d}{dx}(\log x \cdot \log(\sin x)) = \log(\sin x) \cdot \frac{1}{x} + \log x \cdot \frac{d}{dx}(\log(\sin x)) \]

### Step 5: Differentiate \( \log(\sin x) \)
To differentiate \( \log(\sin x) \), we use the chain rule:
\[ \frac{d}{dx}(\log(\sin x)) = \frac{1}{\sin x} \cdot \cos x = \cot x \]
Thus, we have:
\[ \frac{d}{dx}(\log x \cdot \log(\sin x)) = \log(\sin x) \cdot \frac{1}{x} + \log x \cdot \cot x \]

### Step 6: Substitute back into the equation
Now substituting back into our differentiated equation:
\[ \frac{1}{y} \frac{dy}{dx} = \frac{\log(\sin x)}{x} + \log x \cdot \cot x \]

### Step 7: Solve for \( \frac{dy}{dx} \)
Multiplying both sides by \( y \) to isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = y \left( \frac{\log(\sin x)}{x} + \log x \cdot \cot x \right) \]

### Step 8: Substitute \( y \) back
Since \( y = (\sin x)^{\log x} \), we can substitute back:
\[ \frac{dy}{dx} = (\sin x)^{\log x} \left( \frac{\log(\sin x)}{x} + \log x \cdot \cot x \right) \]

### Final Answer
Thus, the derivative is:
\[ \frac{dy}{dx} = (\sin x)^{\log x} \left( \frac{\log(\sin x)}{x} + \log x \cdot \cot x \right) \]