Calculus - Differentials and Partial Derivatives

[Combinatorics]

Homework Statement

Find a differential of second order of a function $u=f(x,y)$ with continuous partial derivatives up to third order at least.Hint: Take a look at $du$ as a function of the variables $x$, $y$, $dx$, $dy$:
$du= F(x,y,dx,dy)=u_xdx +u_ydy$.

Homework Equations

The Attempt at a Solution

I'll be glad to receive some guidance regarding the following:
1) Does the second order differential is assumed to be:
$d^2 f = ( \frac{ \partial}{ \partial x_1} \Delta x_1 +\frac{ \partial}{ \partial x_2} \Delta x_2 ) ^n f$ ?
2) If so, how should I solve this question? Thanks in advance !

 

[bigplanet401]

You know what du is, so what's $$d^2 u = d(du) = ?$$

 

[HallsofIvy]

df= f_xdx+ f_ydy.

As bigplanet401 said, you want the differential of that- but you also need to know that differentials are skew commutative. That is, that dxdy= -dydx.

 

[Combinatorics]

Thanks a lot !