Chain Rule, second partials.

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Homework Statement



I'm curious to know if I'm actually doing this correctly.

Suppose f(x,y) is a function where x = p(s,t) and y = g(s,t) so that w(s,t) = f(x,y).

Compute ws and then wst

Homework Equations



Chain Rule.

The Attempt at a Solution



So! Lets compute ws first. Whenever I use a subscript I refer to the partial with respect to that variable.

[itex]w_s = f_x x_s + f_y y_s[/itex]

That was nice and easy... now for the hard part that I'm not sure of. Lets compute wst :

[itex]w_{st} = (f_{xx} x_t + f_{xy} y_t)x_s + f_x x_{st} + (f_{xy} x_t + f_{yy} y_t)y_s + f_y y_{st}[/itex]

I think this is correct, but with all the variables floating around I'm not entirely sure I didn't miss anything. I would appreciate it very much if someone could verify this for me as I don't want to have a 'blah' moment when my exam happens.
 

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