SOLUTION: I need this for work ASAP. What is the derivative {{{d(f(x) ^ g(x))/dx}}} I need this for work ASAP. What is the derivative d/dx( f(x) ^ g(x) ) wh

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: I need this for work ASAP. What is the derivative {{{d(f(x) ^ g(x))/dx}}} I need this for work ASAP. What is the derivative d/dx( f(x) ^ g(x) ) wh      Log On


   

Question 39439: I need this for work ASAP.
What is the derivative
d%28f%28x%29+%5E+g%28x%29%29%2Fdx
I need this for work ASAP.
What is the derivative
d/dx( f(x) ^ g(x) )
where ^ is the sign of EXPONENTIATION and f and g are function of x.
Please NOTE AGAIN that I am not talking about a compound function, I am talking about one function taken TO THE DEGREE of another.
thanks
Found 2 solutions by longjonsilver, astromathman:
Answer by longjonsilver(2297) About Me  (Show Source):
You can put this solution on YOUR website!
have a read of this i found on google:

http://www.physicsforums.com/showthread.php?t=87992

basically... take natural logs first then differentiate the whole thing.

Good luck :-)
Jon.

Answer by astromathman(21) About Me  (Show Source):
You can put this solution on YOUR website!
We don't have a commonly used theorem for this kind of operation (such as the product rule, etc.).
We can get the g(x) out of the exponent if we take the ln of the function, but we have to undo the damage by taking the exponential (i.e. the inverse of ln) of the result. Note that exp(ln x)=x is normally written like this:
e%5E%28ln%28x%29%29=x
Following this pattern we can write:
f%28x%29%5Eg%28x%29 = e%5E%28ln%28f%28x%29%5Eg%28x%29%29%29 = e%5E%28g%28x%29%2Aln%28f%28x%29%29%29
---
Now we can differentiate using the chain rule and product rule:
---
d%28e%5E%28g%28x%29%2Aln%28f%28x%29%29%29%29%2Fdx =