Multivariable Calculus: Chain Rule and Second Derivatives

[theBEAST]

Homework Statement

Here is the problem with the solution:
http://dl.dropbox.com/u/64325990/MATH%20253/Capture.PNG

What I don't understand is circled in red, how did they combine dxdy with dydx? Is it with Clairaut's theorem? If it is can someone explain how it works in this case because the second derivative makes this very confusing. Additionally 4s^2 ≠ 4r^2 so how can you use Clairaut?

 

[gopher_p]

That's the "By continuity of partials". There's a theorem regarding continuity of partials and the equality of mixed partials. I'm sure it's in your text. You should check it out. It's kind of a big deal.

Yes it's Clairaut.