Q: Derivative of function where independent variable appears as a power as well. 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? 

A: 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.
