Differentiate a^ (x^(x^2)) wrt x where a is a positive constant?

:=

y = a·(x^x²)

Take the natural log of each side in preparation for logarithmic differentiation:

ln y = ln [a·(x^x²)]

ln y = ln a + ln (x^x²)

ln y = ln a + x²·lnx

Now take the derivative implicitly wrt x:

y'/y = 2xlnx + x²/x

y'/y = 2xlnx + x

y'/y = x(2lnx + 1)

y'/y = x(lnx² + 1)

y' = y·x(lnx² + 1)

or

y' = a·(x^x²)·x(lnx² + 1) = a·(x^(x²+1))(lnx² + 1)