`(d)/(dx)(x^(x))` is equal to

Options

A. `x^(x)log(e//x)`
B. `x^(x)logex`
C. `logex`
D. `x^(x)logx`

Answer

To find the derivative of the function \( y = x^x \), we can follow these steps:

### Step-by-Step Solution:

1. **Take the Natural Logarithm**:
We start by taking the natural logarithm of both sides:
\[ \ln(y) = \ln(x^x) \]

2. **Use Logarithmic Properties**:
Using the property of logarithms, \( \ln(a^b) = b \ln(a) \), we can simplify the right side:
\[ \ln(y) = x \ln(x) \]

3. **Differentiate Both Sides**:
Now, we differentiate both sides with respect to \( x \). Remember to use implicit differentiation on the left side:
\[ \frac{d}{dx}(\ln(y)) = \frac{d}{dx}(x \ln(x)) \]
The left side becomes:
\[ \frac{1}{y} \frac{dy}{dx} \]
For the right side, we apply the product rule:
\[ \frac{d}{dx}(x \ln(x)) = \ln(x) + 1 \]

4. **Set Up the Equation**:
Now we have:
\[ \frac{1}{y} \frac{dy}{dx} = \ln(x) + 1 \]

5. **Solve for \(\frac{dy}{dx}\)**:
Multiply both sides by \( y \) to isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = y(\ln(x) + 1) \]

6. **Substitute Back for \( y \)**:
Recall that \( y = x^x \):
\[ \frac{dy}{dx} = x^x(\ln(x) + 1) \]

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