Thinking of sequence where $f_n'$ does not converge to $f'$ everyone,
we see in Rudin this example.

I was trying to think of another example that satisfies this property, but I could not.  I could think of sequences of functions that were similar in form with a manipulation of $\sin(nx)$ on the top but could not think of anything else that satisfies the property/example above.  Is there any examples satisfiying the property above not of the form $\sin(nx)$? Thank you.

Consider the sequence of functions $f_n(x) = \frac{x}{1+nx^2}$. We have that $f'_n(x) = \frac{1-nx^2}{(1+nx^2)^2}$. Now obviously we have that $f(x) = 0$ and $f'(x) = 0$. On the other side we have that
$$\lim_{n \to \infty} f'_n(x) = \begin{cases}
0, & x \not = 0 \\
1, & x = 0
\end{cases}
$$
So the two "derivatives" doesn't coincide at $0$. This has something to do with the fact that $f'_n(x)$ doesn't converge uniformly to $f'(x)$ on any interval contaning $0$.
On the other side it's possible for $g'_n(x)$ not to converge uniformly on an interval and yet $\lim_{n \to \infty} g_n'(x) = g'(x)$ on it. Such an example is $g_n(x) = \frac{e^{-n^2x^2}}{n}$, with $g'_n(x)$  not converging uniformly on an interval containing the origin, yet $\lim_{n \to \infty} g_n'(x) = 0  = g'(x)$ on $\mathbb{R}$
