$g_x\left(x\,y\right)\=G_x\left(x\,y\right)\=2x\sin \left(\frac{1}{x^2\+y^2}\right)-\frac{2x\cos \left(\frac{1}{x^2\+y^2}\right)}{x^2\+y^2}$ 

$g_y\left(x\,y\right)\=G_y\left(x\,y\right)\=2y\sin \left(\frac{1}{x^2\+y^2}\right)-\frac{2y\cos \left(\frac{1}{x^2\+y^2}\right)}{x^2\+y^2}$

$g_x\left(0\,0\right)\=\underset{h\to 0}{lim}\frac{G\left(0\+h\,0\right)-0}{h}\=\underset{h\to 0}{lim}\frac{h^2 \sin \left(\frac{1}{h^2}\right)}{h}\=0$

$g_y\left(0\,0\right)\=\underset{k\to 0}{lim}\frac{G\left(0\,0\+k\right)-0}{k}\=\underset{k\to 0}{lim}\frac{k^2 \sin \left(\frac{1}{k^2}\right)}{k}\=0$

Initialize

$G\left(x\,y\right)\=\left(x^2\+y^2\right)\sin \left(\frac{1}{x^2\+y^2}\right)$$\stackrel{\text{assign as function}}{\to }$$G$

Obtain the first partial derivatives for $\left(x\,y\right)\ne \left(0\,0\right)$ 

$\frac{\partial }{\partial x} G\left(x\,y\right)$ = $2x\sin \left(\frac{1}{x^2\+y^2}\right)-\frac{2x\cos \left(\frac{1}{x^2\+y^2}\right)}{x^2\+y^2}$

$\frac{\partial }{\partial y} G\left(x\,y\right)$ = $2y\sin \left(\frac{1}{x^2\+y^2}\right)-\frac{2y\cos \left(\frac{1}{x^2\+y^2}\right)}{x^2\+y^2}$

Obtain $g_x\left(0\,0\right)$ and $g_y\left(0\,0\right)$ 

$\underset{h\to 0}{lim}\frac{G\left(0\+h\,0\right)-0}{h}$ = $0$

$\underset{k\to 0}{lim}\frac{G\left(0\,0\+k\right)-0}{k}$ = $0$

Obtain $g_x\left(0\,0\right)$ and $g_y\left(0\,0\right)$ from first principles

$\frac{G\left(h\,0\right)}{h}$ = $h\sin \left(\frac{1}{h^2}\right)$$\stackrel{\text{limit}}{\to }$$0$

$\frac{G\left(0\,k\right)}{k}$ = $k\sin \left(\frac{1}{k^2}\right)$$\stackrel{\text{limit}}{\to }$$0$

Initialize

$G≔\left(x\,y\right)\to \left(x^2\+y^2\right)\sin \left(\frac{1}{x^2\+y^2}\right)\:$

Obtain the first partial derivatives for $\left(x\,y\right)\ne \left(0\,0\right)$

$\mathrm{diff}\left(G\left(x\,y\right)\,x\right)$ = $2x\sin \left(\frac{1}{x^2\+y^2}\right)-\frac{2x\cos \left(\frac{1}{x^2\+y^2}\right)}{x^2\+y^2}$

$\mathrm{diff}\left(G\left(x\,y\right)\,y\right)$ = $2y\sin \left(\frac{1}{x^2\+y^2}\right)-\frac{2y\cos \left(\frac{1}{x^2\+y^2}\right)}{x^2\+y^2}$

Obtain $g_x\left(0\,0\right)$ and $g_y\left(0\,0\right)$

$A≔\frac{G\left(0\+h\,0\right)-0}{h}$

$h\sin \left(\frac{1}{h^2}\right)$

$\mathrm{limit}\left(A\,h\=0\right)$ = $0$

$B≔\frac{G\left(0\,0\+k\right)-0}{k}$

$k\sin \left(\frac{1}{k^2}\right)$

$\mathrm{limit}\left(B\,k\=0\right)$ = $0$