If `y=x^(x)`, what is `(dy)/(dx)` at x = 1 equal to?

Options

A. 0
B. 1
C. `-1`
D. 2

Explanation

To find the derivative \(\frac{dy}{dx}\) of the function \(y = x^x\) at \(x = 1\), we will follow these steps:

### Step 1: Take the natural logarithm of both sides
We start with the equation:
\[ y = x^x \]
Taking the natural logarithm on both sides gives:
\[ \ln y = \ln(x^x) \]
Using the property of logarithms, we can simplify the right side:
\[ \ln y = x \ln x \]

### Step 2: Differentiate both sides with respect to \(x\)
Now we differentiate both sides. Remember that when we differentiate \(\ln y\), we use the chain rule:
\[ \frac{d}{dx}(\ln y) = \frac{1}{y} \frac{dy}{dx} \]
For the right side, we differentiate \(x \ln x\) using the product rule:
\[ \frac{d}{dx}(x \ln x) = \ln x + 1 \]
Thus, we have:
\[ \frac{1}{y} \frac{dy}{dx} = \ln x + 1 \]

### Step 3: Solve for \(\frac{dy}{dx}\)
Now, we can solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = y(\ln x + 1) \]
Since \(y = x^x\), we substitute back:
\[ \frac{dy}{dx} = x^x (\ln x + 1) \]

### Step 4: Evaluate \(\frac{dy}{dx}\) at \(x = 1\)
Now, we need to evaluate \(\frac{dy}{dx}\) at \(x = 1\):
\[ \frac{dy}{dx} \bigg|_{x=1} = 1^1 (\ln 1 + 1) \]
Since \(\ln 1 = 0\), we have:
\[ \frac{dy}{dx} \bigg|_{x=1} = 1 (0 + 1) = 1 \]

### Final Answer
Thus, \(\frac{dy}{dx}\) at \(x = 1\) is equal to:
\[ \boxed{1} \]