This maybe a very stupid question. I forgot a lot of things about calculus :(
$F(x, y)$ is an differentiable function. Anybody could explain me why I could get the following equation ?
$$F(x+yd\phi, y+xd\phi) = F(x,y) + (y\frac{\partial{F}}{\partial{x}} + x\frac{\partial{F}}{\partial{y}})d\phi.$$
I think that there is some confusions in the notations you used. Furthermore, is the equation missing some terms and symbols ?
There are some problems in the equation as stated currently : $$ F(x+yd\phi, y+xd\phi) = F(x,y) + (y\frac{\partial{F}}{\partial{x}} + x\frac{\partial{F}}{\partial{y}})d\phi $$
For example, $F(x+yd\phi, y+xd\phi)$ is a number, while $\frac{\partial{F}}{\partial{y}}$ is a function.
I'm guessing that the result you want is only the definition of a differentiable function : $$ F(a+h_1, b+h_2) = F(a,b) + d_{(a,b)}F \cdot (h_1,h_2) + o(||(h_1,h_2)||) $$
Applied here with $(a,b) = (x,y)$ and $(h_1,h_2) = (yd\phi, xd\phi)$, we get : $$ F(x+yd\phi, y+xd\phi) = F(x,y) + d_{(x,y)}F \cdot (yd\phi, xd\phi) + o(||(yd\phi, xd\phi)||) $$ Recalling that : $$ d_{(x,y)}F \cdot (yd\phi, xd\phi) = \frac{\partial{F}}{\partial{x}}(x,y) \cdot yd\phi + \frac{\partial{F}}{\partial{y}}(x,y) \cdot xd\phi $$ We finally get : $$ F(x+yd\phi, y+xd\phi) = F(x,y) + \left(y\frac{\partial{F}}{\partial{x}}(x,y) + x\frac{\partial{F}}{\partial{y}}(x,y)\right) d\phi + o(||(yd\phi, xd\phi)||) $$
Which is what I understood your equation meant.