multivariate computation – $ F (x, y) = int_ {f (x, y)} ^ {g (x, y)} h (x, y, t) dt $. Find $ frac { partial F} { partial x} $.

$ f, g: mathbb {R} ^ 2 to mathbb {R} $ and $ h: mathbb {R} ^ 3 to mathbb {R} $ be $ C ^ 1 $, to define $ F (x, y) = int_ {f (x, y)} ^ {g (x, y)} h (x, y, t) dt $. Find $ frac { partial F} { partial x} $

So I separated $ F (x, y) = int_0 ^ {g (x, y)} h (x, y, t) dt- int_0 ^ {f (x, y)} h (x, y, t) dt $

And then using FTC, I think I have $ F (x, y) = H (x, y, g (x, y)) – H (x, y, 0) -H (x, y, f (x, y)) + H (x, y) , 0) = H (x, y, g (x, y)) – H (x, y, f (x, y)) $

Then differentiate this new thing from x.

$ frac { partial F} { partial x} = frac { partial H} { partial x} + frac { partial H} { partial y} + frac { partial H} { partial g (x, y)} – frac { partial H} { partial x} – frac { partial H} { partial y} – frac { partial H} { partial f (x, y) } $

Is it correct? From there, I would simplify the expression and develop the partial ones. But I'm not sure that the FTC works that way for $ H $since it is a function of 3 variables.