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

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

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

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

Initialize

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

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

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

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

Obtain $f_x\left(0\,0\right)$ and $f_y\left(0\,0\right)$ 

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

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

Obtain $f_x\left(0\,0\right)$ and $f_y\left(0\,0\right)$ from first principles

$\frac{F\left(h\,0\right)}{h}$ = $h\sin \left(\frac{1}{\sqrt{h^2}}\right)$$\stackrel{\text{assuming real}}{\to }$$h\sin \left(\frac{1}{\left|h\right|}\right)$$\stackrel{\text{limit}}{\to }$$0$$$

$\frac{F\left(0\,k\right)}{k}$ = $k\sin \left(\frac{1}{\sqrt{k^2}}\right)$$\stackrel{\text{assuming real}}{\to }$$k\sin \left(\frac{1}{\left|k\right|}\right)$$\stackrel{\text{limit}}{\to }$$0$$$

Initialize

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

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

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

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

Obtain $f_x\left(0\,0\right)$ and $f_y\left(0\,0\right)$

$A≔\mathrm{simplify}\left(\frac{F\left(0\+h\,0\right)-0}{h}\right) assuming \mathrm{real}$

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

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

$B≔\mathrm{simplify}\left(\frac{F\left(0\,0\+k\right)-0}{k}\right) assuming \mathrm{real}$

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

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