Is this differentiation under the integral justified by Leibniz rule?

Question

I am trying to understand the following proof from Hunter's "Applied Analysis":

I was wondering why we need to bring up the dominated convergence theorem to justify differentiation under the integral. We have $\varphi$ is an element of the Schwarz space which is infinitely differentiable so we just use the Leibniz rule to justify this right?

Edit: This is the theorem for differentiating under integral