Is taking the derivative inside this integral acceptable?

Question

I have run into an integral where I take the partial derivative of x when the upper limit of the integral is x. This is how I solve it but I am not sure if this is correct. Can someone please check? Thank you! $$ \frac{\partial}{\partial x}\int_0^xf(t)dt=\int_0^x\frac{\partial f(t)}{\partial x}dt=0 $$

Answer 1

No.

Assuming that $x$ does not appear free in $f(t)$, then by the fundamental theorem of calculus:

$$\dfrac{\partial ~~}{\partial x}\int_0^x f(t)\mathrm d t=f(x)$$