How to differentiate y= x^(e^x)

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