Let $f_n(x) = \frac{x^n}{n}, 0 \leq x \leq1$. Then
$1) $ the sequence $f_n$ converge uniformly to a function $f$ on $[0,1]$.
$2)$ the sequence $f_n '$ converge to a function $g$ on $[0,1]$ and $f'(x) = g(x), x \in (0,1), f'(1) \neq g'(1)$.
I have done the first part. i.e. the sequence $f_n$ converge uniformly to a function $f(x) = 0$ on $[0,1]$. I am stuck with the second part.
Here $f_n '(x) = x^{n-1}$ Thus $lim_{n \to \infty} x^{n-1} = 0$.
If $x=1$, then $f'_n(x) = 1 \to_{n \to \infty} 1$.
If $0$ $\le$ $ x < 1$, then $f'_n(x) \to_{n \to \infty} 0$.
Thus:
$$f'_n(x) \to g(x) = \begin{cases} 0, \text{if $0\le x < 1$} \\ 1, \text{if $x = 1$} \end{cases}$$