Differentiation of complex Powers? ie 2^x^2?

Question

I always have trouble remembering how to differentiate complex powers.

x^2 is easy! Its 2x!

But what happens when you have more powers than that?

What if you have 2^x^2 or 2^x^3^x ???

How do you handle these things?

Answer 1

y = 2^(x²)
log y = (x²) (log 2)
(1/y) (dy/dx) = (2) (log 2) (x)
(dy/dx) = (y) (2) (log 2) (x)
dy/dx = (2^x²)(2) (log 2) (x)
dy/dx = 2^(x² + 1) (log 2) (x)

Answer 2

consider 2^x^2=y
log 2^x^2=log y
x^2*log 2=log y
(1/y)dy/dx=log 2*(2*x) [log 2 is a constant]
dy/dx=(2^x^2) *(log 2)(2*x)
this is the ans