Posted by AHKSIJ on Wednesday, December 7, 2011 at 12:08pm.
f = g(e^2x)
f' = 2e^2x g'
f'' = 4e^2xg' + 2e^2xg''
example:
g(x) = x sinx
f(x) = g(e^2x) = e^2x sin(e^2x)
g' = sinx + xcosx
g'' = cosx + cosx - xsinx
= 2cosx - xsinx
f' = 2e^2x sin(e^2x) + e^2xcos(e^2x)*2e^2x
= 2e^2x(sin(e^2x) + e^2x cos(e^2x))
Letting g(e^2x) = g(u)
f' = 2e^2x g'(u)
I have expanded out
4e^2xg'(u) + 2e^2xg''(u)
and it checks out. Have lots of blank paper!
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