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Derivative of a function to a function 
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#1
Oct1707, 09:31 AM

P: 44

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
Oct1707, 09:37 AM

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P: 2,481

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
Oct1707, 11:04 AM

P: 44

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
Oct1707, 12:35 PM

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P: 2,919

Derivative of a function to a function



#5
Oct1707, 01:34 PM

Mentor
P: 15,167

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



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