Find `(dy)/(dx)fory=x^xdot`

Explanation

To find \(\frac{dy}{dx}\) for the function \(y = x^x\), we can follow these steps:

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

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

### Step 3: Differentiate both sides with respect to \(x\)
Now, we differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}(\ln y) = \frac{d}{dx}(x \ln x) \]

Using the chain rule on the left side:
\[ \frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(x \ln x) \]

### Step 4: Differentiate the right-hand side using the product rule
To differentiate \(x \ln x\), we use the product rule:
\[ \frac{d}{dx}(x \ln x) = \ln x + x \cdot \frac{1}{x} = \ln x + 1 \]

### Step 5: Substitute back into the equation
Now we substitute this back into our equation:
\[ \frac{1}{y} \frac{dy}{dx} = \ln x + 1 \]

### Step 6: Solve for \(\frac{dy}{dx}\)
Multiplying both sides by \(y\) gives:
\[ \frac{dy}{dx} = y(\ln x + 1) \]

### Step 7: Substitute \(y\) back in
Since we know that \(y = x^x\), we substitute this back into the equation:
\[ \frac{dy}{dx} = x^x(\ln x + 1) \]

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

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