y= x^(e^x)

To differentiate, first we will apply the natural logarithm to both sides:

==> lny = ln [x^(e^x)]

We know that: ln a^b = b*ln a

==> lny = (e^x) * ln x

Now we will differentiate both sides:

==> (lny)' = [e^x)*lnx]'

To differentiate e^x * ln x we will use the product rule:

[(e^x)*lnx]' = (e^x)'*lnx + (e^x)*(lnx)'

              = (e^x)lnx + e^x *1/x

==> (1/y) y' = (e^x)*lnx + e^x(1/x)

==> (y'/y) =( e^x)( lnx/x)

==> y' = y*(e^x)*lnx /x