Real Analysis question comparing functions I am trying to compare the functions $\lim f'_n$ and $(\lim f_n)'$with the following sequence,
$$f_n(x)=\frac{x^n}{n}$$
for $x\in [0,1]$
So I have already $limf_n'$ by calculating the derivative with respect to x and got the following,
$$f_n'(x) = x^{n-1}$$
So $f_n'(x)$ converges pointwise to 0 if x is in $[0,1)$ and $1$ if $x=1$
I believe I have done this part correct and now I am trying to compare it to $(\lim f_n)'$ but I am not quite sure how to calculate that

Note that 
$$\lim_{n\to \infty}f_n(x)=\lim_{n\to \infty}\frac{x^n}{n}=0$$
for all $x\in [0,1]$.  
What is the derivative of $0$?
The point of the exercise is to show that the derivative of the limit of a sequence of functions (i.e, $(\lim_{n\to \infty}f_n(x))'$) is not always equal to the limit of the derivative of that sequence of functions (i.e., $\lim_{n\to \infty} f_n'(x)$).
There are sufficient conditions (uniform convergence of the sequence of derivatives,  $f_n'(x)$) for which they will be equal, but those are not met here.
