What is an example of a sequence of functions, $(f_n)_{n=1}^\infty\subset C^1([a,b])$, which are bounded in $C^1([a,b])$ under $\|\cdot\|_{\infty}$ but are such that their first derivatives $\|f'_n\|_{\infty}\to\infty$ as $n\to\infty$?
There are many examples on this site for a single function, but I am looking for an intance when we have a sequence of functions. Are there any particularly simple instances which can be provided to demonstrate the above point?
Take $f_n(x)=\sin(nx)$ as per the comment of Mindlack.
The sequence $(\sin(nx))_{n=1}^\infty\subset C^1([a,b])$ is bounded since for all $n\in\mathbb N$ one has that $\|\sin(nx)\|_{\infty}=1$.
Additionally, for each $n\in\mathbb N$ we have that $f_n'(x)=n\cos(nx)$. Then, $$\|f_n'\|_\infty=\|n\cos(nx)\|_\infty=|n|\|\cos(nx)\|_\infty= n,$$ which implies that $\lim_{n\to\infty}\|f_n'\|_\infty=\infty$. Thus, the derivatives of $(\sin(nx))_{n=1}^\infty\subset C^1([a,b])$ are unbounded.