Chain rule of a multivariate function

Question

I was doing some exercises of the chain rule in multivariate calculus, but i saw this problem and i'm not sure where to start, if you could give me some advice.

Let $f$ differential on $(u,v)$ and $g$ a function of $(x,y)$ given $g(x,y)=f(x^2-y^2,2xy)$ compute $g_x$ and $g_y$ in terms of $f_x$ and $f_y$

Answer 1

we have $f(u,v)$ with $u(x,y)=x^2-y^2$, $v(x,y)=2xy$ thus

$$g_x=f_u u_x+ f_v v_x=f_u \cdot 2x+f_v \cdot2y$$

$$g_y=f_u u_y+ f_v v_y=f_u \cdot -2y+f_v \cdot2x$$