Prove, for an open $\Omega \subset \mathbb{R}^n$ with $x\in \Omega$, that $u\in W^{1,p}(\Omega-\{x\})\implies u\in W^{1,p}(\Omega).$

Question

Let $n\geq 2$, $\Omega\subset \mathbb{R}^n$ open, $x\in \Omega$ and $p\geq1$ I want to prove the above implication. We just need to show that the weak derivative of $u$ on the punctured domain remains the weak derivative on $\Omega$. I am unsure of how to do this though. I was thinking we could take a function $\phi \in C_0^{\infty}(\Omega)$ with $x\in supp(\phi)$ and define a new sequence of functions by smoothly cutting $\phi$ off in a ball of decreasing radius around $x$ and then pass to the limit in the definition. However we need $L^1$ convergence of the derivatives but I think this fails.