Is taking the derivative inside this integral acceptable? 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!
$$ \frac{\partial}{\partial x}\int_0^xf(t)dt=\int_0^x\frac{\partial f(t)}{\partial x}dt=0 $$
No.
Assuming that $x$ does not appear free in $f(t)$, then by the fundamental theorem of calculus:
$$\dfrac{\partial ~~}{\partial x}\int_0^x f(t)\mathrm d t=f(x)$$
No.
Assuming that $x$ does not appear free in $f(t)$, then by the fundamental theorem of calculus:
$$\dfrac{\partial ~~}{\partial x}\int_0^x f(t)\mathrm d t=f(x)$$