Derivative of a function to a function

1. Oct 17, 2007

joex444

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. Oct 17, 2007

EnumaElish

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)].

3. Oct 17, 2007

joex444

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.

4. Oct 17, 2007

nrqed

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 $\ln(g(x))$ and then to isolate g'(x).

5. Oct 17, 2007

D H

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