Differentiate the following w.r.t. x :
`(x)^(logx)`

Answer

To differentiate the function \( y = x^{\log x} \) with respect to \( x \), 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 differentiation process:
\[ \log y = \log(x^{\log x}) \]

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

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

### Step 4: Solve for \(\frac{dy}{dx}\)
Now, we can isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = y \cdot \frac{2 \log x}{x} \]

### Step 5: Substitute back for \(y\)
Recall that \( y = x^{\log x} \):
\[ \frac{dy}{dx} = x^{\log x} \cdot \frac{2 \log x}{x} \]

### Step 6: Simplify the expression
We can simplify the expression:
\[ \frac{dy}{dx} = 2 \log x \cdot x^{\log x - 1} \]

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

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