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

Options

A. 1
B. log x
C. log (ex)
D. None

Answer

To solve the problem where \( y = \log(x^x) \) and we need to find \( \frac{dy}{dx} \), we can follow these steps:

### Step 1: Simplify the Expression
Using the logarithmic property \( \log(a^b) = b \cdot \log(a) \), we can rewrite \( y \):
\[ y = \log(x^x) = x \cdot \log(x) \]

### Step 2: Differentiate Using the Product Rule
Now we need to differentiate \( y \) with respect to \( x \). Since \( y = u \cdot v \) where \( u = x \) and \( v = \log(x) \), we apply the product rule:
\[ \frac{dy}{dx} = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx} \]
Here, \( \frac{du}{dx} = 1 \) and \( \frac{dv}{dx} = \frac{1}{x} \).

### Step 3: Substitute and Simplify
Substituting \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the product rule gives:
\[ \frac{dy}{dx} = x \cdot \frac{1}{x} + \log(x) \cdot 1 \]
This simplifies to:
\[ \frac{dy}{dx} = 1 + \log(x) \]

### Step 4: Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 1 + \log(x) \]