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.

Another example is the sequence of functions
$$\begin{array}{l|rcl}
f_n : & [-1,1] & \longrightarrow & \mathbb{R} \\
    & x & \longmapsto & \sqrt{\frac{1}{n^2}+x^2} \end{array}$$ These functions are differentiable according to the chain rule as $g_n(x): x \longmapsto \frac{1}{n^2}+x^2$ is strictly positive on $[-1,1]$. We also have $$x^2 \le f_n^2(x) = \frac{1}{n^2}+x^2 \le (\vert x \vert +\frac{1}{n})^2$$ hence
    $$\vert x \vert \le f_n(x) \le \vert x \vert +\frac{1}{n}$$ as $f_n$ is positive. So by the squeeze test, $(f_n)$ converges uniformly to the absolute value function $f(x) = \vert x \vert$. But this function is not differentiable at $0$. Thus, the uniform limit of differentiable functions need not be differentiable.
The picture below helps you to understand graphically what is happening:

And for more details, you can go to my counterexamples website.
