# 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.