Let $f \in L^1(\mathbb R)$ and $\phi \in C_c^{\infty} (\mathbb R)$. I am pretty certain that the following is true:
$$\lim_{h \to 0} \int_{\mathbb R} f(x) \frac{\phi(x-h) - \phi(x)}{h} dx = - \int_{\mathbb R} f(x) \phi'(x) dx$$
However, I can't prove it. I want to use dominated convergence, but I can't seem to find a dominating function. Any help?
At first we observe that $\max\{|\phi^{'}(x)|:x\in\mathbb{R}\}<\infty$. This is true because $\phi\in C_{c}^{\infty}$. Now fix $h>0$, by mean value theorem we have: $$\frac{\phi(x-h)-\phi(x)}{h}=-\phi^{'}(y_{h})$$ for some $y_{h}\in (x-h,x)$, for negative $h$ you can do the same but with $y_h\in(x,x-h).$ Finally, as you suggested we use dominated convergence theorem $$|f(x)\frac{\phi(x-h)-\phi(x)}{h}|=|f(x)|\cdot|\phi^{'}(y_{h})|\leq |f(x)|\cdot \max\{|\phi^{'}(x)|:x\in\mathbb{R}\}.$$ Observing that $|f(x)|\cdot \max\{|\phi^{'}(x)|:x\in\mathbb{R}\}\in L^1(\mathbb{R})$ we obtain the desired equality.