How do you differentiate y=x^x?

To solve this problem, you need to put it into simplest form which is putting it into natural logarithm both the RHS and LHS. Then differentiate both side with respect to x as shown below.

In(y) = ln(x^x)    - natural logarithm both side

ln(y) = xln(x)      - using the power rule

(1/y)dy/dx = x*1/x + ln(x)      - Diffrentiate both side (chain rule in the RHS) 

dy/dx = y(1+ln(x))     - multiplying  'y' in both sides

         = x^x(1+ln(x))    - replacing the value of 'y'