Vaguely, it is the kind of error I would expect of AI, not of W|A. However, the Notebook Assistant gives a correct derivation.

Here is what I got the second time I tried (the first one was shorter):

To find the derivative of x^x, we can use logarithmic differentiation. Here are the steps:

1. Rewrite the expression using logarithms:

   Start by taking the natural logarithm of both sides of the equation y==x^x:

   ln(y)==ln(x^x)==xln(x)
2. **Differentiate both sides with respect to x:**

   Using implicit differentiation, differentiate both sides:

   d/(d x) [ln(y)]==d/(d x) [xln(x)]
   The left side differentiates to:

   1/y (d y)/(d x)
   The right side, using the product rule, differentiates to:

   ln(x)+x\[CenterDot]1/x==ln(x)+1
3. **Solve for (d y)/(d x):**

   Multiply both sides by y to solve for the derivative:

   (d y)/(d x)==y(ln(x)+1)
4. **Substitute back y==x^x:**

   Finally, replace y with x^x to get the derivative in terms of x:

   (d y)/(d x)==x^x (ln(x)+1)
So, the derivative of x^x is: 
x^x(ln(x)+1)
This matches the result you have in your notebook. If you need any more help, feel free to ask!

Yes, this is exactly how I invoked the Wolfram|Alpha step-by-step solution.