If ` y=x^(logx) ,then (dy)/(dx)=`

Options

A. ` 2x ^((logx-1)) logx `
B. ` x^((logx-1))`
C. ` (2)/( xlogx )`
D. ` x^((logx-1))logx `

Answer

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

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

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

### Step 3: Differentiate both sides
Now we differentiate both sides with respect to \( x \). We will use the chain rule on the left side and the product rule on the right side:
\[ \frac{d}{dx}(\log y) = \frac{1}{y} \frac{dy}{dx} \]
For the right side, we differentiate \( (\log x)^2 \):
\[ \frac{d}{dx}((\log x)^2) = 2 \log x \cdot \frac{d}{dx}(\log x) = 2 \log x \cdot \frac{1}{x} \]

### Step 4: Set the derivatives equal
Now we equate the derivatives:
\[ \frac{1}{y} \frac{dy}{dx} = \frac{2 \log x}{x} \]

### Step 5: Solve for \( \frac{dy}{dx} \)
To isolate \( \frac{dy}{dx} \), we multiply both sides by \( y \):
\[ \frac{dy}{dx} = y \cdot \frac{2 \log x}{x} \]

### Step 6: Substitute back for \( y \)
Recall that \( y = x^{\log x} \), so we substitute this back into our equation:
\[ \frac{dy}{dx} = x^{\log x} \cdot \frac{2 \log x}{x} \]

### Step 7: Simplify the expression
We can simplify the expression:
\[ \frac{dy}{dx} = 2 x^{\log x - 1} \log x \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 2 x^{\log x - 1} \log x \]

---