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 #