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Posted by on Wednesday, December 7, 2011 at 12:08pm.

Calculate f" for f(x)=g(e^(2x)), where g is a function defined for all real numbers & g admits second order derivative.

  • calculus - , Wednesday, December 7, 2011 at 1:35pm

    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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