Partial Differentiation 2

Sorry, i'm stuck, here goes....

$\displaystyle u=ax+by$

$\displaystyle v=bx-ay$

In the first two parts of the question, you are asked to show that:

$\displaystyle \frac{\partial u}{\partial x})_{y}.\frac{\partial x}{\partial u})_{v}=\frac{a^{2}}{a^{2}+b^{2}} $

and

$\displaystyle \frac{\partial y}{\partial v})_{x}.\frac{\partial v}{\partial y})_{u}=\frac{a^{2}+b^{2}}{a^{2}} $

which i have done, and it's all fine and dandy :)

However, the last part confuzes me.... :(

If $\displaystyle f(x,y)=f(u,v)$, show that:

$\displaystyle \frac{\partial^{2} f}{\partial x \partial y}=ab (\frac{\partial^{2} f}{\partial u^{2}} - \frac{\partial^{2} f}{\partial v^{2}}) + (b^{2}-a^{2})\frac{\partial^{2} f}{\partial u \partial v} $

The problem is i'm not sure what $\displaystyle f(x,y)=f(u,v)$actually means. I understand it to be: f is a function of two variables, x and y, and f is a function of two variables, u and v, and these two functions are equal. But u and v are themselves functions of x and y so what is f?

A push in the right direction would be greatly appreciated! :)