Partial

at $\left(x\,y\right)$ 

at $\left(a\,b\right)$ 

$f_{\mathrm{xx}}$

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

$8$

$f_{\mathrm{xy}}$

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

$10$

$f_{\mathrm{yx}}$

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

$10$

$f_{\mathrm{yy}}$

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

$12$

Table 4.2.2(a)   Second partials of $f$ at $\left(x\,y\right)$ and $\left(-3\,2\right)$ 

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

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

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

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

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

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

$\mathrm{f\_\_xy}\left(x\,y\right)$ = $\frac{2\left(x-y\right)}{{\left(x\+y\right)}^3}$$$

$\mathrm{f\_\_yx}\left(x\,y\right)$ = $\frac{2\left(x-y\right)}{{\left(x\+y\right)}^3}$

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

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

$f≔\frac{x-y}{x\+y}\:$

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

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

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

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

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

$8$

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

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

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

$10$

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

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

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

$10$

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

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

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

$12$

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

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

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

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

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

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

$8$

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

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

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

$10$

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

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

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

$10$

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

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

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

$12$