Differentiating an integral using dominated convergence
By the mean value theorem, there is, for each $t$, a $c_t(x) \in (0,1)$ with
$$\frac{f(x+h,t) - f(x,t)}{h} = \frac{\partial f}{\partial x}(x + c_t(x)\cdot h, t).$$
So if the partial derivative $\frac{\partial f}{\partial x}$ is locally uniformly (in $x$) dominated by $g$, then so are the difference quotients. And that makes the dominated convergence theorem applicable.