Well, first let's make it look nicer...
y = 2^(3^(x^2))
Let color(darkred)(u = 3^(x^2))
=> y = 2^u
Using logarithmic simplification:
ln_2(y) = u -> (lny)/(ln2) = u -> lny = u ln2
And now with some manipulation:
y = e^(u ln2) = e^(ln2^u) = 2^u
color(green)((dy)/(du) = e^(u ln2)*ln2 = 2^u ln2)
Similarly:
Let color(darkred)(w = x^2)
=> u = 3^w
ln_3(u) = w -> (lnu)/(ln3) = w -> lnu = wln3
u = e^(wln3) = e^(ln3^w) = 3^w
color(green)((du)/(dw) = e^(wln3)*ln3 = 3^w ln3)
Lastly, color(green)((dw)/(dx) = 2x).
Thus, taking the whole derivative:
d/(dx)[y = 2^(u(w(x)))] = d/(dx)[y = 2^(u(x^2))] = d/(dx)[y = 2^(3^(x^2))] = ((dy)/(du)) ((du)/(dw)) ((dw)/(dx))
= (2^u ln2)(3^w ln3)(2x)
= (2^(3^(x^2)) ln2)(3^(x^2) ln3)(2x)
= 2xln2ln3(2^(3^(x^2))3^(x^2))
= color(blue)(xln2ln3(2^(3^(x^2) + 1)3^(x^2)))
