When does $\frac{\partial}{\partial t}\int_0^x f(x') dx'=x \frac{\partial f}{\partial t}$?

Question

Under what conditions do the following relation holds? $$\frac{\partial}{\partial t}\int_0^x f(x') dx'=x \frac{\partial f}{\partial t}$$ Should it be stated that $f(0)=0$?

Let's say that I know the primitive function of $f$ when integrated on $x$: $\int f(x) dx=F(x)$. Then, I have $$\frac{\partial}{\partial t}\int_0^x f(x') dx'= \frac{\partial F(x)}{\partial t}-\frac{\partial F(0)}{\partial t}$$ which I don't know how to compute.

Answer 1

As written, the integral on the left does not involve $t$, so the LHS is identically zero. So the RHS must also be identically zero, and $f$ must be constant.

Answer 2

This is just an application of Leibnitz Rule. It is sometimes also known as Differentiation under the Integral sign.

Using this, you can easily obtain the following:

$$\frac{\partial}{\partial t}\int_0^x f(x') dx'= \frac{\partial F(x)}{\partial t}-\frac{\partial F(0)}{\partial t}$$