By definition, a random variable $X$ has density $f_X$, if
$$P(\mathbf{a}\leq X\leq \mathbf{b}, t)=\int _{\mathbf{a}}^{\mathbf{b}}f_{X}(\mathbf{x}, t)\,\mathrm{d}\mathbf{x}$$
Now, for this integral to be well-defined is necessary to have at least $f_X(\mathbf{x}, t)\in L^1(\mathbb{R}^n\times\mathbb{R}^+)$. But suppose I want to compute a derivative of $f_X$. It is all right if the derivative operator is applied after the integral, but can it be brought inside? I'm afraid it can't be done. As an example, any truncated Weierstrass function is integrable, but it remains nowhere differentiable. So how is one supposed to do it?