Let $f \in W^{1,p}(\mathbb{R}^{n})$, where $p \in (1,\infty)$. Let us define $f^{i}_{h}$ as
$$ f^{i}_{h}(x) := \frac{f(x+he_{i}) - f(x)}{|h|}. $$
Prove that
$$ ||f^{i}_{h} - D_{i}f||_{L^{p}(\mathbb{R}^{n})} \to 0. $$
I don't have any idea how to prove that. I've just heard about 7.11 from Trudinger's book about elliptic equations, but I don't have any concludes from that lemmas. Maybe it's a simple fact, but I will be really grateful for help, proof or source, where can I find that.