Partial

at $\left(x\,y\right)$ 

at $\left(a\,b\right)$ 

$f_{\mathrm{xx}}$

$\frac{2xy\left(x^2-3y^2\right)}{{\left(x^2\+y^2\right)}^3}$

$-\frac{276}{2197}$

$f_{\mathrm{xy}}$

$-\frac{x^4-6x^2{y}^2\+y^4}{{\left(x^2\+y^2\right)}^3}$

$\frac{119}{2197}$

$f_{\mathrm{yx}}$

$-\frac{x^4-6x^2{y}^2\+y^4}{{\left(x^2\+y^2\right)}^3}$

$\frac{119}{2197}$

$f_{\mathrm{yy}}$

$-\frac{2xy\left(3x^2-y^2\right)}{{\left(x^2\+y^2\right)}^3}$

$-\frac{36}{2197}$

Table 4.2.1(a)   Second partials of $f$ at $\left(x\,y\right)$ and $\left(2\,3\right)$ 

$f\=\frac{x y}{x^2\+y^2}$$\stackrel{\text{assign}}{\to }$

Second partial derivatives at $\left(x\,y\right)$ 

$\frac{{\partial }^2}{{\partial }^{}{x}^2} f$ = $-\frac{6xy}{{\left(x^2\+y^2\right)}^2}\+\frac{8x^{3}y}{{\left(x^2\+y^2\right)}^3}$$\stackrel{\text{simplify}}{\=}$$\frac{2xy\left(x^2-3y^2\right)}{{\left(x^2\+y^2\right)}^3}$$$

$\frac{{\partial }^2}{\partial y\partial x} f$ = $\frac{1}{x^2\+y^2}-\frac{2x^2}{{\left(x^2\+y^2\right)}^2}-\frac{2y^2}{{\left(x^2\+y^2\right)}^2}\+\frac{8x^2{y}^2}{{\left(x^2\+y^2\right)}^3}$$\stackrel{\text{simplify}}{\=}$$-\frac{x^4-6x^2{y}^2\+y^4}{{\left(x^2\+y^2\right)}^3}$$$

$\frac{{\partial }^2}{\partial x\partial y} f$ = $\frac{1}{x^2\+y^2}-\frac{2x^2}{{\left(x^2\+y^2\right)}^2}-\frac{2y^2}{{\left(x^2\+y^2\right)}^2}\+\frac{8x^2{y}^2}{{\left(x^2\+y^2\right)}^3}$$\stackrel{\text{simplify}}{\=}$$-\frac{x^4-6x^2{y}^2\+y^4}{{\left(x^2\+y^2\right)}^3}$$$

$\frac{{\partial }^2}{{\partial }^{}{y}^2} f$ = $-\frac{6xy}{{\left(x^2\+y^2\right)}^2}\+\frac{8x{y}^3}{{\left(x^2\+y^2\right)}^3}$$\stackrel{\text{simplify}}{\=}$$-\frac{2xy\left(3x^2-y^2\right)}{{\left(x^2\+y^2\right)}^3}$$$

Second partial derivatives at $\left(a\,b\right)$ 

$\mathrm{f\_\_xx}\left(x\,y\right)\=\frac{{\partial }^2}{{\partial }^{}{x}^2} f$$\stackrel{\text{assign as function}}{\to }$$\mathrm{f\_\_xx}$

$\mathrm{f\_\_xy}\left(x\,y\right)\=\frac{{\partial }^2}{\partial y\partial x} f$$\stackrel{\text{assign as function}}{\to }$$\mathrm{f\_\_xy}$

$\mathrm{f\_\_yy}\left(x\,y\right)\=\frac{{\partial }^2}{{\partial }^{}{y}^2} f$$\stackrel{\text{assign as function}}{\to }$$\mathrm{f\_\_yy}$

$\mathrm{f\_\_yx}\left(x\,y\right)\=\frac{{\partial }^2}{\partial x\partial y} f$$\stackrel{\text{assign as function}}{\to }$$\mathrm{f\_\_yx}$

Obtain the second partial derivatives at $\left(x\,y\right)$ 

$\mathrm{f\_\_xx}\left(x\,y\right)$ = $-\frac{6yx}{{\left(x^2\+y^2\right)}^2}\+\frac{8x^{3}y}{{\left(x^2\+y^2\right)}^3}$$\stackrel{\text{simplify}}{\=}$$\frac{2xy\left(x^2-3y^2\right)}{{\left(x^2\+y^2\right)}^3}$$$

$\mathrm{f\_\_xy}\left(x\,y\right)$ = $\frac{1}{x^2\+y^2}-\frac{2x^2}{{\left(x^2\+y^2\right)}^2}-\frac{2y^2}{{\left(x^2\+y^2\right)}^2}\+\frac{8x^2{y}^2}{{\left(x^2\+y^2\right)}^3}$$\stackrel{\text{simplify}}{\=}$$-\frac{x^4-6x^2{y}^2\+y^4}{{\left(x^2\+y^2\right)}^3}$$$

$\mathrm{f\_\_yx}\left(x\,y\right)$ = $\frac{1}{x^2\+y^2}-\frac{2x^2}{{\left(x^2\+y^2\right)}^2}-\frac{2y^2}{{\left(x^2\+y^2\right)}^2}\+\frac{8x^2{y}^2}{{\left(x^2\+y^2\right)}^3}$$\stackrel{\text{simplify}}{\=}$$-\frac{x^4-6x^2{y}^2\+y^4}{{\left(x^2\+y^2\right)}^3}$

$\mathrm{f\_\_yy}\left(x\,y\right)$ = $-\frac{6yx}{{\left(x^2\+y^2\right)}^2}\+\frac{8x{y}^3}{{\left(x^2\+y^2\right)}^3}$$\stackrel{\text{simplify}}{\=}$$-\frac{2xy\left(3x^2-y^2\right)}{{\left(x^2\+y^2\right)}^3}$$$

Obtain the second partial derivatives at $\left(a\,b\right)$ 

$f≔\frac{x y}{x^2\+y^2}\:$

Second partials at $\left(x\,y\right)$ 

Second partials at $\left(a\,b\right)$ 

$\mathrm{f\_\_xx}≔\mathrm{simplify}\left(\mathrm{diff}\left(f\,x\,x\right)\right)$

$\frac{2xy\left(x^2-3y^2\right)}{{\left(x^2\+y^2\right)}^3}$

$\mathrm{eval}\left(\mathrm{f\_\_xx}\,\left[x\=2\,y\=3\right]\right)$

$-\frac{276}{2197}$

$\mathrm{f\_\_xy}≔\mathrm{simplify}\left(\mathrm{diff}\left(f\,x\,y\right)\right)$

$-\frac{x^4-6x^2{y}^2\+y^4}{{\left(x^2\+y^2\right)}^3}$

$\mathrm{eval}\left(\mathrm{f\_\_xy}\,\left[x\=2\,y\=3\right]\right)$

$\frac{119}{2197}$

$\mathrm{f\_\_yx}≔\mathrm{simplify}\left(\mathrm{diff}\left(f\,y\,x\right)\right)$

$-\frac{x^4-6x^2{y}^2\+y^4}{{\left(x^2\+y^2\right)}^3}$

$\mathrm{eval}\left(\mathrm{f\_\_yx}\,\left[x\=2\,y\=3\right]\right)$

$\frac{119}{2197}$

$\mathrm{f\_\_yy}≔\mathrm{simplify}\left(\mathrm{diff}\left(f\,y\,y\right)\right)$

$-\frac{2xy\left(3x^2-y^2\right)}{{\left(x^2\+y^2\right)}^3}$

$\mathrm{eval}\left(\mathrm{f\_\_yy}\,\left[x\=2\,y\=3\right]\right)$

$-\frac{36}{2197}$

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

Second partials at $\left(x\,y\right)$ 

Second partials at $\left(a\,b\right)$ 

$\mathrm{simplify}\left({\mathrm{D}}_{1\,1}\left(f\right)\left(x\,y\right)\right)$

$\frac{2xy\left(x^2-3y^2\right)}{{\left(x^2\+y^2\right)}^3}$

${\mathrm{D}}_{1\,1}\left(f\right)\left(2\,3\right)$

$-\frac{276}{2197}$

$\mathrm{simplify}\left({\mathrm{D}}_{1\,2}\left(f\right)\left(x\,y\right)\right)$

$-\frac{x^4-6x^2{y}^2\+y^4}{{\left(x^2\+y^2\right)}^3}$

${\mathrm{D}}_{1\,2}\left(f\right)\left(2\,3\right)$

$\frac{119}{2197}$

$\mathrm{simplify}\left({\mathrm{D}}_{2\,1}\left(f\right)\left(x\,y\right)\right)$

$-\frac{x^4-6x^2{y}^2\+y^4}{{\left(x^2\+y^2\right)}^3}$

${\mathrm{D}}_{2\,1}\left(f\right)\left(2\,3\right)$

$\frac{119}{2197}$

$\mathrm{simplify}\left({\mathrm{D}}_{2\,2}\left(f\right)\left(x\,y\right)\right)$

$-\frac{2xy\left(3x^2-y^2\right)}{{\left(x^2\+y^2\right)}^3}$

${\mathrm{D}}_{2\,2}\left(f\right)\left(2\,3\right)$

$-\frac{36}{2197}$