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

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Answer 1 · 2016-04-26T14:56:26

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

$2^(2x) = (2^2)^x = 4^x$

$d/dx(a^x) = a^x lna$ $" "$ (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

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

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