$F_x$

$\=G_u u_x\+G_v v_x$

$F_y$

$\=G_u u_y\+G_v v_y$

$\=G_u \left(2 x\right)\+G_v \left(2 y\right)$

$\=G_u \left(-2 y\right)\+G_v \left(2 x\right)$

$\=2\left(x G_u\+y G_v\right)$

$\=2\left(x G_v-y G_u\right)$

Obtain $F_x$ 

$\frac{\partial }{\partial x} G\left(x^2-y^2\,2 x y\right)$ = $2{\mathrm{D}}_1\left(G\right)\left(x^2-y^2\,2xy\right)x\+2{\mathrm{D}}_2\left(G\right)\left(x^2-y^2\,2xy\right)y$$$$$

Obtain $F_y$ 

$\frac{\partial }{\partial y} G\left(x^2-y^2\,2 x y\right)$ = $-2{\mathrm{D}}_1\left(G\right)\left(x^2-y^2\,2xy\right)y\+2{\mathrm{D}}_2\left(G\right)\left(x^2-y^2\,2xy\right)x$$$$$

Initialize

$$\begin{gathered} \mathrm{interface}\left(\mathrm{typesetting}\=\mathrm{extended}\right)\: \\ \mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[u\left(x\,y\right)\,v\left(x\,y\right)\right]\right)\: \\ \mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}\=\mathrm{true}\right)\: \end{gathered}$$

Maple's representation of $F_x$ and $F_y$  

$\mathrm{diff}\left(G\left(u\,v\right)\,x\right)$

${\mathrm{D}}_1\left(G\right)\left(u\,v\right)\left(u_x\right)+{\mathrm{D}}_2\left(G\right)\left(u\,v\right)\left(v_x\right)$

$\mathrm{diff}\left(G\left(u\,v\right)\,y\right)$

${\mathrm{D}}_1\left(G\right)\left(u\,v\right)\left(u_y\right)+{\mathrm{D}}_2\left(G\right)\left(u\,v\right)\left(v_y\right)$

Implement the chain rule

$\mathrm{Typesetting}:-\mathrm{Unsuppress}\left(\left[u\left(x\,y\right)\,v\left(x\,y\right)\right]\right)\:$

$$\begin{gathered} u≔x^2-y^2\: \\ v≔2 x y\: \end{gathered}$$$$

$\mathrm{diff}\left(G\left(u\,v\right)\,x\right)$ = $2{\mathrm{D}}_1\left(G\right)\left(x^2-y^2\,2xy\right)x+2{\mathrm{D}}_2\left(G\right)\left(x^2-y^2\,2xy\right)y$$$

$\mathrm{diff}\left(G\left(u\,v\right)\,y\right)$ = $-2{\mathrm{D}}_1\left(G\right)\left(x^2-y^2\,2xy\right)y+2{\mathrm{D}}_2\left(G\right)\left(x^2-y^2\,2xy\right)x$$$