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I have $f(x)=(1+\frac{1}{x})^{x}$, I need to find the derivative of this, using the definition of derivative, and show that f is monotonically increasing.

Using the definition, I have that $\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h} = \lim_{h\rightarrow0}\frac{(1+\frac{1}{x+h})^{x+h}-(1+\frac{1}{x})^{x}}{h}$ whose numerator is the same as $(1+\frac{1}{x+h})^{x}(1+\frac{1}{x+h})^{h}-(1+\frac{1}{x})^{x} $

At this point, I'm not sure how to proceed. Would I need to use binomial formula?

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    `show that f is monotonically increasing`... Not interested anymore?2012-12-01

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For this kind of mess, it is nice to first know the correct answer. The logarithmic derivative seems indicated.

$\ln f(x) = x \ln(1+1/x) = x \ln((x+1)/x) = x \ln(x+1) - x \ln(x)$, so $f'(x)/f(x) = x/(x+1) + \ln(x+1) - 1 - \ln(x) = -1/(x+1) + \ln(1+1/x) $. This tells you that you should get two terms in your result and one of them should involve $\ln(1+1/x)$.

As to how you should continue to get this, I am not sure.