I have an integral
$$ \frac{\partial}{\partial x} \int_{0}^{t} f(x(\tau),\tau) d\tau, $$
where $f=f(x,t)$ is a function of a dependent variable $x=x(t)$ and the independent variable $t$ (time).
Can I let the derivative enter under the integral sign? Under what conditions can I do that, and why?
Update: The motivation behind wanting to insert the partial derivative inside the integral before integrating is that I know from other facts (say, physics) that $\partial f/\partial x$ is another function $g(x,t)$ that is much simpler to integrate directly than $f$. Indeed, the form of $f$ may be unknown but the partial derivative of it is known. For example in canonical transformations and Hamiltonian mechanics, $f$ could be a Hamiltonian $H(x,y;t)$ whose partial derivative $\partial H/\partial x=d y/dt$, making the integral exact and the result simply $\Delta y=y(t)-y(0)$. But I want to know if such tactic is rigorous.