After literally hours of "no, and what and how", I need help with the following task:

f(u,v)= g(x(u,v),y(u,v))

x(u,v)= u^2- v^2

y(u,v)= 2uv

Expressing f'u and f'v as partial derivatives of g I get:

f'u= (dg/dx)*(dx/du) + (dg/dy)*(dy/du)

f'v= (dg/dx)*(dx/dv) + (dg/dy)*(dy/dv)

(written in crooked d's)

Now I'm supposed to find all functions f that satisfy u*f'u - v*f'v=0

The hardest part (which confuse me the most because there are so many steps involving the chain rule is the following): "Express f''uu+ f''vv through partial derivatives of g. Simplify as much as possible"

Please help me with either of the tasks, I need this by monday and I'm so confused and tired of staring at this task!