Derivative

at $\left(x\,y\right)$ 

at $\left(3\,-2\right)$ 

$f_{\mathrm{xxx}}$

$-\frac{12y}{{\left(x-y\right)}^4}$

$\frac{24}{625}$

$f_{\mathrm{xxy}}\=f_{\mathrm{xyx}}\=f_{\mathrm{yxx}}$

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

$-\frac{4}{625}$

$f_{\mathrm{xyy}}\=f_{\mathrm{yxy}}\=f_{\mathrm{yyx}}$

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

$-\frac{16}{625}$

$f_{\mathrm{yyy}}$

$\frac{12x}{{\left(x-y\right)}^4}$

$\frac{36}{625}$

Table 4.2.6(a)   The distinct third-order partial derivatives of $f$

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

Obtain the distinct third-order partial derivatives at $\left(x\,y\right)$ and at $\left(3\,-2\right)$

$\frac{{\partial }^3}{{\partial }^{}{x}^3} f$ = $\frac{6}{{\left(x-y\right)}^3}-\frac{6\left(x\+y\right)}{{\left(x-y\right)}^4}$$\stackrel{\text{simplify}}{\=}$$-\frac{12y}{{\left(x-y\right)}^4}$$$$\stackrel{\text{evaluate at point}}{\to }$$\frac{24}{625}$

$\frac{{\partial }^3}{{\partial }^{} x^2\partial y} f$ = $-\frac{2}{{\left(x-y\right)}^3}\+\frac{6\left(x\+y\right)}{{\left(x-y\right)}^4}$$\stackrel{\text{simplify}}{\=}$$\frac{4\left(x\+2y\right)}{{\left(x-y\right)}^4}$$$$\stackrel{\text{evaluate at point}}{\to }$$-\frac{4}{625}$

$\frac{{\partial }^3}{\partial x\partial y^2} f$ = $-\frac{2}{{\left(x-y\right)}^3}-\frac{6\left(x\+y\right)}{{\left(x-y\right)}^4}$$\stackrel{\text{simplify}}{\=}$$-\frac{4\left(2x\+y\right)}{{\left(x-y\right)}^4}$$$$\stackrel{\text{evaluate at point}}{\to }$$-\frac{16}{625}$

$\frac{{\partial }^3}{{\partial }^{}{y}^3} f$ = $\frac{6}{{\left(x-y\right)}^3}\+\frac{6\left(x\+y\right)}{{\left(x-y\right)}^4}$$\stackrel{\text{simplify}}{\=}$$\frac{12x}{{\left(x-y\right)}^4}$$\stackrel{\text{evaluate at point}}{\to }$$\frac{36}{625}$$$$$

Define the substitution equations for P:$\left(3\,-2\right)$

$P\=\left[x\=3\,y\=-2\right]$$\stackrel{\text{assign}}{\to }$

Obtain the distinct third-order partial derivatives at P:$\left(3\,-2\right)$

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

Apply the D-operator to obtain the distinct third-order partial derivatives

${\mathrm{D}}_{1\,1\,1}\left(f\right)\left(3\,-2\right)$ = $\frac{24}{625}$$$

${\mathrm{D}}_{1\,1\,2}\left(f\right)\left(3\,-2\right)$ = $-\frac{4}{625}$$$

${\mathrm{D}}_{1\,2\,2}\left(f\right)\left(3\,-2\right)$ = $-\frac{16}{625}$$$

${\mathrm{D}}_{2\,2\,2}\left(f\right)\left(3\,-2\right)$ = $\frac{36}{625}$$$