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Question
I have found the derivative of this function i asked about earlier but now i can't seem to find a way to solve for the zeroes because i cant isolate x after simplifying the derivative.
f(x) = (2009^x)/(x^2009) where x > 0
The simplified derivative that i got was:
f'(x) = 2009^x[ln2009(x^2009) + 2009x^2008]
The derivative is the quotient rule where f(x) = g(x)/h(x) where
g(x) = 2009^x and h(x) = x^2009.
g'(x) = ln(2009)2009^x and h'(x) = 2009x^2008.
The derivative is (lo d hi - hi d lo)/loČ. This gives us
f'(x) = (x^2009*ln(2009)2009^x - 2009^x*2009x^2008)/x^4018.
Note that 2008^x can be taken out of each term in the numerator and denominator, giving
f'(x) = (x*ln(2009)2009^x - 2009^x*2009)/x^2010.
Factoring out 2009^x gives f'(x) = 2009^x(x*ln(2009) - 2009)/x^2010.
That's what I get, and you're close.
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