Techniques

If ${\left\lvert {{\frac{\partial }{\partial t}\,}f(x, t)} \right\rvert} \leq g(x) \in L^1$, then letting $F(t) = \int f(x, t) ~dt$, $$\begin{align*} {\frac{\partial }{\partial t}\,} F(t) &\coloneqq\lim_{h \rightarrow 0} \int \frac{f(x, t+h)-f(x, t)}{h} d x \\ &\mathop{\mathrm{=}}^{\scriptstyle\text{DCT}} \int {\frac{\partial }{\partial t}\,} f(x, t) ~dx .\end{align*}$$

To justify passing the limit, let $h_k \to 0$ be any sequence and define $$\begin{align*} f_k(x, t) = \frac{f(x, t+h_k)-f(x, t)}{h_k} ,\end{align*}$$ so $f_k \overset{k\to\infty}\longrightarrow{\frac{\partial f}{\partial t}\,}$ pointwise.

Apply the MVT to $f_k$ to get $f_k(x, t) = f_k(\xi, t)$ for some $\xi \in [0, h_k]$, and show that $f_k(\xi, t) \in L_1$.