suppose i have F(x,y), and i want to evaluate d2F. is the following correct?

$\displaystyle dF = \frac{{\partial F}}{{\partial x}}dx + \frac{{\partial F}}{{\partial y}}dy$

$\displaystyle d^2F = \frac{{\partial}}{{\partial x}} dF + \frac{{\partial}}{{\partial y}}dF$

$\displaystyle \frac{{\partial}}{{\partial x}} dF = \frac{{\partial^2 F}}{{\partial x^2}}dx^2 + \frac{{\partial^2F}}{{\partial y \partial x}}dydx$

$\displaystyle \frac{{\partial}}{{\partial y}} dF = \frac{{\partial^2 F}}{{\partial y^2}}dy^2 + \frac{{\partial^2F}}{{ \partial x \partial y}}dxdy$

$\displaystyle \frac{{\partial^2F}}{{\partial x \partial y}} dxdy = \frac{{\partial^2F}}{{\partial y \partial x}} dydx \Rightarrow$

$\displaystyle d^2F = \frac{{\partial^2 F}}{{\partial x^2}} dx^2 + \frac{{\partial^2F}}{{\partial y^2}} dy^2 + 2 \frac{{\partial^2 F}}{{\partial y \partial x}}dydx$

i don't believe i ever studied higher-order total derivatives in college, and now that i have a need to make use of it, i am unsure if $\displaystyle 2 \frac{{\partial^2 F}}{{\partial y \partial x}}dydx$ should be $\displaystyle \frac{{\partial^2 F}}{{\partial y \partial x}}dydx$

thanks in advance!

edit:

thanks for the clue-in on LaTeX plato.