Update: Someone gives me a good counterexample, which basically answers all the questions I posed. The example is, $$f_n(x)=\frac{sin(nx)}{n}$$ －－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－
I just came up with these basic questions:
Let $\Omega$ be an open subset of $\mathbb{R}^n$. If $f_n \rightarrow f$ in $ L^1(\Omega)$, and $f_n'$ exists $a.e$ in $\Omega$, then is it true that $f'$ exists $a.e.$ in $\Omega$ and $f_n' \rightarrow f' a.e. $? What about a subsequence? （This question can be answered by thinking about Weirestrass Theorem without thinking about a specific counterexample! Thanks to my friend's comment.)
If the previous statement is false, then one may ask another question: If $f_n \rightarrow f$ in $ L^1(\Omega)$, $f_n'$, $f'$ exists $a.e.$ in $\Omega$, then is true that $f_n' \rightarrow f' a.e. $? What about a subsequence? (This question has been answered by $Jose27. Thanks!)
What if we have $f_n'$ is uniformly bounded? Or more weakly, $f_n'$ uniformly bounded in $L^1$?
Generally, what condition on $f_n'$ can guarantee $f'$ exists $a.e.$ and $f_n' \rightarrow f' a.e.$ in $\Omega$? Suppose $f'$ exists $a.e.$, what conditions on $f_n'$ can guarantee $f_n' \rightarrow f' a.e.$?
The motivation of this kind of question is I'm studying "changing the order of limits" phenomena. For example, define $\delta_h g(x)=\frac{g(x+h)-g(x)}{h}$, then $$\lim_{h \rightarrow 0} \delta_h f_n(x) \rightarrow f_n'(x)$$ Notice that for fixed $h$, we have $$\lim_{n \rightarrow \infty} \delta_h f_n(x)=\delta_hf(x)$$ Hence $$\lim_{h \rightarrow 0} \lim_{n \rightarrow \infty} \delta_h f_n(x)=\lim_{h \rightarrow 0}\delta_hf(x)$$ If we can change the order of limit, then we have $$f_n' \rightarrow f' $$
The problem is that changing the order is subtle, but can the almost eveywhere convergence get rid of the subtlety? How about choosing a subsequence?
Thanks for @Jose27's comments. The example shows that even any subsequence does not work without some bounded conditions on $f_n'$. But I want to understand more.
Any comments and partial ideas would be fully appreciated!
By the way, another motivation for the questions above comes from my research on the trace of BV functions. I've seen a lot of cases dealing with change of the order of limits. Although for the above questions, it may be not hard to find some simple examples, but understanding what really going on there would be of much help if thinking some deeper problems. Thinking this kind of problem would not be in vain.