`y=(log_(e)x)^(sinx)`

Answer

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

### Step 1: Take the natural logarithm of both sides
We start by taking the natural logarithm of both sides to simplify the expression.

\[ \log y = \log((\log_e x)^{\sin x}) \]

### Step 2: Apply the power rule of logarithms
Using the property of logarithms that states \( \log(a^b) = b \cdot \log a \), we can rewrite the equation:

\[ \log y = \sin x \cdot \log(\log_e x) \]

### Step 3: Differentiate both sides with respect to \( x \)
Now, we differentiate both sides with respect to \( x \). We will use implicit differentiation on the left side and the product rule on the right side.

\[ \frac{d}{dx}(\log y) = \frac{d}{dx}(\sin x \cdot \log(\log_e x)) \]

Using the chain rule on the left side:

\[ \frac{1}{y} \frac{dy}{dx} \]

For the right side, we apply the product rule:

\[ \frac{d}{dx}(\sin x) \cdot \log(\log_e x) + \sin x \cdot \frac{d}{dx}(\log(\log_e x)) \]

### Step 4: Differentiate the components
Now we differentiate each component:

1. The derivative of \( \sin x \) is \( \cos x \).
2. For \( \log(\log_e x) \), we use the chain rule:
\[ \frac{d}{dx}(\log(\log_e x)) = \frac{1}{\log_e x} \cdot \frac{1}{x} = \frac{1}{x \log_e x} \]

### Step 5: Substitute back into the equation
Substituting these derivatives back into our equation gives:

\[ \frac{1}{y} \frac{dy}{dx} = \cos x \cdot \log(\log_e x) + \sin x \cdot \frac{1}{x \log_e x} \]

### Step 6: Solve for \( \frac{dy}{dx} \)
Now, we can solve for \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = y \left( \cos x \cdot \log(\log_e x) + \frac{\sin x}{x \log_e x} \right) \]

### Step 7: Substitute \( y \) back
Recall that \( y = (\log_e x)^{\sin x} \), we substitute this back in:

\[ \frac{dy}{dx} = (\log_e x)^{\sin x} \left( \cos x \cdot \log(\log_e x) + \frac{\sin x}{x \log_e x} \right) \]

### Final Answer
Thus, the derivative of \( y = (\log_e x)^{\sin x} \) is:

\[ \frac{dy}{dx} = (\log_e x)^{\sin x} \left( \cos x \cdot \log(\log_e x) + \frac{\sin x}{x \log_e x} \right) \]

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