Prove Leibniz' rule in $H^1$.

Question

Using that, in dimension $1$, the Hilbert space $H^1(I)\subset C^{0,1/2}(\overline{I})$ I want prove that $(fg)'=f'g+fg'$, $f,g\in H^1(I)$ without using density arguments. I have been able to prove it if $f\in H^1(I)$ and $g\in C^1(I)$ with compact support. Any hint?