$f_n\xrightarrow{L^p}f \implies f_n'\xrightarrow{L^p}f'$?

Question

Let $(f_n)_{n\ge 1}$ be a sequence of function that is differentiable. such that $f_n\xrightarrow{L^p}f$ converges. I was wondering if $f_n'\xrightarrow{L^p}f'$. So far I could only find this result but it doesn't really apply here I think.

Answer 1

No.

Example: $p=1$, $f_n$ and $f$ in $L^1[0,1]$ let be given by

$$ f_n(x) =x^n , \quad f(x)=0.$$

Then

$$||f_n-f||_1 = \frac{1}{n+1} \to 0,$$

as $ n \to \infty,$ but

$$||f_n'-f'||_1 = 1$$

for all $n$.