Suppose $F : (0,\infty) \times [0,1] \to \mathbb{R}$ is defined by $$F(t,y) := \int_{0}^{y} f(t,x)~dx$$ and additionally $$\partial_{t}f(t,x) = \partial_{x}g(t,x),$$ where $f,g$ are smooth functions on $(0,\infty) \times [0,1]$.
Question: is this the correct evaluation of $\partial_{t}F(t,y)$?
$$\partial_{t}F(t,y) = \int_{0}^{y} \partial_{t}f(t,x)~dx = \int_{0}^{y} \partial_{x}g(t,x)~dx = g(t,y) - g(t,0)$$ where we make use of the FTC to achieve the final equality.
All correct. In particular, that final step is valid because the FTC says that $$\int_a^b\partial_xg(t,x)\,\mathrm dx=g(t,b)-g(t,a)$$ provided that $g$ is continuously differentiable on $[a,b]$ with respect to $x.$