$f_x\=\{\begin{array}{cc}\frac{y\left(x^4\+4x^2{y}^2-y^4\right)}{{\left(x^2\+y^2\right)}^2}& \left(x\,y\right)\ne \left(0\,0\right)\\ 0& \left(x\,y\right)\=\left(0\,0\right)\end{array}$

$f_y\=\{\begin{array}{cc}\frac{x\left(x^4-4x^2{y}^2-y^4\right)}{{\left(x^2\+y^2\right)}^2}& \left(x\,y\right)\ne \left(0\,0\right)\\ 0& \left(x\,y\right)\=\left(0\,0\right)\end{array}$

Table 4.2.5(a)   First partial derivatives of $f\left(x\,y\right)$.

$f_{\mathrm{xx}}\=\{\begin{array}{cc}-\frac{4x{y}^3\left(x^2-3y^2\right)}{{\left(x^2\+y^2\right)}^3}& \left(x\,y\right)\ne \left(0\,0\right)\\ 0& \left(x\,y\right)\=\left(0\,0\right)\end{array}$

$f_{\mathrm{xy}}\=\{\begin{array}{cc}\frac{x^6\+9x^4{y}^2-9x^2{y}^4-y^6}{{\left(x^2\+y^2\right)}^3}& \left(x\,y\right)\ne \left(0\,0\right)\\ -1& \left(x\,y\right)\=\left(0\,0\right)\end{array}$

$f_{\mathrm{yy}}\=\{\begin{array}{cc}-\frac{4y{x}^3\left(3x^2-y^2\right)}{{\left(x^2\+y^2\right)}^3}& \left(x\,y\right)\ne \left(0\,0\right)\\ 0& \left(x\,y\right)\=\left(0\,0\right)\end{array}$

$f_{\mathrm{yx}}\=\{\begin{array}{cc}\frac{x^6\+9x^4{y}^2-9x^2{y}^4-y^6}{{\left(x^2\+y^2\right)}^3}& \left(x\,y\right)\ne \left(0\,0\right)\\ 1& \left(x\,y\right)\=\left(0\,0\right)\end{array}$

Table 4.2.5(b)   Second-order partial derivatives of $f$ 

$f_{\mathrm{xx}}\left(0\,0\right)\=\underset{h\to 0}{lim}\frac{f_x\left(h\,0\right)-f_x\left(0\,0\right)}{h}\=\underset{h\to 0}{lim}\frac{0}{h}\=0$

$f_{\mathrm{yy}}\left(0\,0\right)\=\underset{k\to 0}{lim}\frac{f_y\left(0\,k\right)-f_y\left(0\,0\right)}{k}\=\underset{k\to 0}{lim}\frac{0}{k}\=0$

$f_{\mathrm{xy}}\left(0\,0\right)\=\underset{k\to 0}{lim}\frac{f_x\left(0\,k\right)-f_x\left(0\,0\right)}{k}\=\underset{k\to 0}{lim}\frac{-k}{k}\= -1$

$f_{\mathrm{yx}}\left(0\,0\right)\=\underset{h\to 0}{lim}\frac{f_y\left(h\,0\right)-f_y\left(0\,0\right)}{h}\=\underset{h\to 0}{lim}\frac{h}{h}\=1$

Table 4.2.5(c)   Second-order partial derivatives at $\left(0\,0\right)$