Derivative of a function to a function

  1. I'm a tutor in physics, but was asked this question: What is the derivative of sin(x)^ln(x), with respect to x?

    I'm not sure how you would go about taking the derivative of a function raised to a function.

    Is there a general for for d/dx ( f(x)^g(x) )?

    I understand the answer involves a ln(sin(x)), according to Maple, and would love to see how you end up with g(f(x)).
  2. jcsd
  3. EnumaElish

    EnumaElish 2,332
    Science Advisor
    Homework Helper

    That's not the LN in your formula. It's a generic LN.

    If g(x) = h(x)^k(x) then g'(x) = g(x) [k(x)h'(x)/h(x) + Log(h(x))k'(x)].
  4. Thanks, that's really neat. Usually we assume k(x) to be a constant, n, so obviously k'(x) would be 0 and the second term drops, leaving us with the power rule.
  5. nrqed

    nrqed 3,103
    Science Advisor
    Homework Helper
    Gold Member

    It must be obvious by now but just in case someone would wonder where the formula provided by Enumaelish comes from, the trick is to not differentiate g(x) itself but to differentiate [itex] \ln(g(x)) [/itex] and then to isolate g'(x).
  6. D H

    Staff: Mentor

    The trick is to express h(x)^k(x) as exp(k(x)*Log(h(x)). Everything follows from that.
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