Differentiate x^x and sketch it

Write y = x^xln(y) = xln(x)1/y dy/dx = (1+ln(x)) by applying the chain rule to the LHS and product rule to the RHSdy/dx = y(1+ln(x)) Rearranging dy/dx = x^x (1+ln(x)) Substituting y=x^x into the equationNote that we cannot sketch y = x^x in general for values of x less than 0, as for every non-integer value of negative x we have to find roots of a negative number which will lie in the complex plane. However we can calculate (for example) (-1)^(-1) = -1 or (-2)^(-2) = 1/4 so it is possible for a few specific values! If you sketch this graph on a computer you will see it plot only a few points for x negative - this is exactly why this happens!