Examples: Sequence of derivatives $\lim_{n\to\infty}f_n'(x)\neq \left(\lim_{n\to\infty}f_n(x)\right)'$

Question

I am looking for some interesting examples of sequences of functions/of derivatives which satisfy the follwoing conditions:

1. $f_n'(x)$ must converge pointwisely,
2. $f_(x)$ should converge uniformly or pointwisely,
3. $\lim_{n\to\infty}f_n'(x)\neq \left(\lim_{n\to\infty}f_n(x)\right)'$.

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So far I have only found:

$f_n:[0,1]\to\mathbb{R}$, where $f_n(x)= \frac{1}{n}x^n$ and $f_n'(x)=x^{n-1}$.

In this case:

$$\left(\lim_{n\to\infty}f_n(x)=\hat{0}\right)'=\hat{0} \neq \lim_{n\to\infty}f_n'(x)= \begin{cases} 1, \text{if } x \ne 1 \\ 0, \text{else. } \end{cases} $$

Maybe someone has some more examples to share :)

EDIT

Maybe it is possible to find an example where the limit of $f_n'(x)$ attains a "nice" form, e.g. a polynomial or an elementary function without the necessity of defining it piecewise where you have jumps.