The value of ` (d)/(dx) (x^x) ` is

Options

A. `x x ^(x-1)`
B. `x^x log ex `
C. `x^x log x `
D. None of these

Answer

To find the derivative of the function \( y = x^x \), we will use logarithmic differentiation. Here’s a step-by-step solution:

### Step 1: Take the natural logarithm of both sides
We start by taking the natural logarithm of both sides of the equation:
\[ \ln y = \ln(x^x) \]

### Step 2: Simplify using logarithmic properties
Using the property of logarithms that states \( \ln(a^b) = b \ln a \), we can simplify the right-hand side:
\[ \ln y = x \ln x \]

### Step 3: Differentiate both sides
Now we differentiate both sides with respect to \( x \). On the left side, we will use the chain rule:
\[ \frac{d}{dx}(\ln y) = \frac{1}{y} \frac{dy}{dx} \]
On the right side, we will use the product rule since \( x \ln x \) is a product of two functions:
\[ \frac{d}{dx}(x \ln x) = \frac{d}{dx}(x) \cdot \ln x + x \cdot \frac{d}{dx}(\ln x) \]
Calculating these derivatives:
- The derivative of \( x \) is \( 1 \).
- The derivative of \( \ln x \) is \( \frac{1}{x} \).

Thus, we have:
\[ \frac{d}{dx}(x \ln x) = 1 \cdot \ln x + x \cdot \frac{1}{x} = \ln x + 1 \]

### Step 4: Set the derivatives equal
Now we set the derivatives from both sides equal to each other:
\[ \frac{1}{y} \frac{dy}{dx} = \ln x + 1 \]

### Step 5: Solve for \( \frac{dy}{dx} \)
To isolate \( \frac{dy}{dx} \), we multiply both sides by \( y \):
\[ \frac{dy}{dx} = y(\ln x + 1) \]

### Step 6: Substitute back for \( y \)
Since we defined \( y = x^x \), we can substitute back:
\[ \frac{dy}{dx} = x^x(\ln x + 1) \]

### Final Answer
Thus, the derivative of \( x^x \) is:
\[ \frac{d}{dx}(x^x) = x^x(\ln x + 1) \]

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