For $x \in \mathbb{R^d}$, why can we take the partial derivative wrt to $x_i$ inside the the following solution
$$\frac{\partial}{\partial x_i}u(x,t) = \int_{\mathbb{R}^d}\frac{\partial}{\partial x_i}\Psi(x-y, t)f(y)dy$$
where $f \in L^{1}(\mathbb{R})$ and $\Psi(\cdot)$ is a general kernel.
I suppose this is because of the Dominated Convergence Theorem, but how do we bound $f$ to apply it, or am I totally off point? For example, if we have the heat kernel, $\Psi(x,t) := \frac{1}{(4t\pi)^{d/2}}\exp(-|x|^2/4t)$, then
$$\frac{1}{(4t\pi)^{d/2}}\exp\left(-|x|^2/4t\right) \leq 1, \quad\text{when}~x>0.$$
Does it make sense to define a function, $g := \sup_{x \in \mathbb{R}^d}f(x) \geq f(x)$, and so now we apply DCT?