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

Options

A. ` log (ex) `
B. ` x^(x) (1+log x ) `
C. ` log ((e)/(x))`
D. ` -log ( (e)/(x))`

Answer

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

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

**Hint:** Remember that logarithmic properties allow you to simplify expressions involving exponents.

### Step 2: Differentiate \( y \) with respect to \( x \)
Now we need to differentiate \( y = x \cdot \log(x) \). This requires the product rule, which states that if \( u = f(x) \) and \( v = g(x) \), then:
\[ \frac{d(uv)}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]
Here, let \( u = x \) and \( v = \log(x) \).

- Differentiate \( u \):
\[ \frac{du}{dx} = 1 \]
- Differentiate \( v \):
\[ \frac{dv}{dx} = \frac{1}{x} \]

Now applying the product rule:
\[ \frac{dy}{dx} = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx} \]
Substituting \( u \) and \( v \):
\[ \frac{dy}{dx} = x \cdot \frac{1}{x} + \log(x) \cdot 1 \]

### Step 3: Simplify the expression
Now simplify the expression:
\[ \frac{dy}{dx} = 1 + \log(x) \]

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

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