Question

How do you differentiate `2^(2x)`?

Answer 1

Unless told that I must use the chain rule, I would not. But see below.

Explanation:

`2^(2x) = (2^2)^x = 4^x`

`d/dx(a^x) = a^x lna` `" "` (A proof of this uses the chain rule.)

So `d/dx(2^(2x)) = d/dx(4^x) = 4^x ln4`

Proof of `d/dx(a^x) = a^x lna`

`(a^x) = (e^ lna)^x = e^(xlna)`

`d/dx(a^x) = d/dx(e^(xlna)) = e^(xlna) d/dx(xlna)`

Note that `lna` is a constant, so that `d/dx(xlna) = lna`.

We continue:

`= e^(xlna) d/dx(xlna) = e^(xlna) lna`

Finally, recall that `e^(xlna) = a^x` (see above), so we finish

`= e^(xlna) lna = a^x lna`