The differentiation process is unstable with the $||f||_{\infty}=\text{max}_{0\leq x\leq 1}|f(x)|$, where $f\in C^1[0,1]$.
I know that the process of differentiating a function is not stable, so take for example $f(x)=1$ and $f_n(x)=1+\frac{1}{n}\sin(n^2x)$. Clearly $f_n\to f$ but $f_n'\nrightarrow f'$. My question is whether changing this normal By $||f||_{C^1[0,1]}=||f||_{\infty}+||f'||_{\infty}$ can we conclude that the differentiation process is stable or unstable? In my opinion it is unstable but I can not find a counterexample, could someone help me please? In the case that it is stable, how would it be tested? Thank you very much.