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Find f’(x) if f(x) = 3^g(x) where g(x) = LOG_3(x^2 - √x + 4/x + 10*LOG_e(x))

  • math -

    if f(x) = 3^g(x)
    then f'(x) = (ln3)(3^(gx)) (g ' (x) )

    g(x) = ???? , do you mean 3 to be the base of LOG
    or are we doing ln ? or what
    Once you have decided, continue as below ...
    so g ' (x) = ....

    insert that result above.

  • math -

    the 3 is the base so log(base3)(...log(base/e)(x)

  • math -

    if g(u) = log_3(u)
    f(x) = 3^g(u) = 3^log_3(u) = u
    so now it's easy:

    f(x) = x^2 - √x + 4/x + 10*ln(x)
    f'(x) = 2x - 1/(2√x) - 4/x^2 + 10/x

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