# Chain Rule, second partials.

1. Oct 13, 2012

### Zondrina

1. The problem statement, all variables and given/known data

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

2. Relevant equations

Chain Rule.

3. 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.

$w_s = f_x x_s + f_y y_s$

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

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

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.