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

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

$\genfrac{}{}{0ex}{}{\frac{\partial f}{\partial x}}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{\left(x\,y\right)\=\left(3\,2\right)}$ =  $4$

$\genfrac{}{}{0ex}{}{\frac{\partial f}{\partial y}}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{\left(x\,y\right)\=\left(3\,2\right)}$=  $9$

Table 4.1.2(a)   First partial derivatives of $f$

Context Panel

$f_x$ 

$f_y$ 

$x y^2-3 y-2$

$x{y}^2-3y-2$

$\stackrel{\text{differentiate w.r.t. x}}{\to }$

$y^2$

$\stackrel{\text{evaluate at point}}{\to }$

$4$

$x y^2-3 y-2$

$x{y}^2-3y-2$

$\stackrel{\text{differentiate w.r.t. y}}{\to }$

$2xy-3$

$\stackrel{\text{evaluate at point}}{\to }$

$9$

Define $f$ as an expression

$x y^2-3 y-2$$\stackrel{\text{assign to a name}}{\to }$$f$

Obtain $f_x\left(x\,y\right)$ and $f_y\left(x\,y\right)$ 

$\frac{\partial }{\partial x} f$ = $y^2$$$

$\frac{\partial }{\partial y} f$ = $2xy-3$$$

Obtain $f_x\left(a\,b\right)$ and $f_y\left(a\,b\right)$

$\genfrac{}{}{0ex}{}{\frac{\partial }{\partial x} f}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=3\,y\=2}$ = $4$$$

$\genfrac{}{}{0ex}{}{\frac{\partial }{\partial y} f}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=3\,y\=2}$ = $9$$$

Define the functions  $f_x$ and $f_y$ 

$f_x$ 

$f_y$ 

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

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

Obtain $f_x\left(x\,y\right)$ and $f_y\left(x\,y\right)$ 

$\mathrm{f\_\_x}\left(x\,y\right)$ = $y^2$$$

$\mathrm{f\_\_y}\left(x\,y\right)$ = $2xy-3$$$

Obtain $f_x\left(a\,b\right)$ and $f_y\left(a\,b\right)$ 

$\mathrm{f\_\_x}\left(3\,2\right)$ = $4$$$

$\mathrm{f\_\_y}\left(3\,2\right)$ = $9$$$

Define $f$ as an expression

$f≔x y^2-3 y-2\:$

Obtain $f_x\left(x\,y\right)$ and $f_y\left(x\,y\right)$ 

$\mathrm{fx}≔\mathrm{diff}\left(f\,x\right)$

$y^2$

$\mathrm{fy}≔\mathrm{diff}\left(f\,y\right)$

$2xy-3$

Obtain $f_x\left(a\,b\right)$ and $f_y\left(a\,b\right)$ 

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

$4$

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

$9$

Define $f$ as a function

$f≔\left(x\,y\right)\to x y^2-3 y-2\:$

Obtain $f_x\left(x\,y\right)$ and $f_y\left(x\,y\right)$ 

${\mathrm{D}}_1\left(f\right)\left(x\,y\right)$ = $y^2$$$

${\mathrm{D}}_2\left(f\right)\left(x\,y\right)$ = $2xy-3$$$

Obtain $f_x\left(a\,b\right)$ and $f_y\left(a\,b\right)$ 

${\mathrm{D}}_1\left(f\right)\left(3\,2\right)$ = $4$$$

${\mathrm{D}}_2\left(f\right)\left(3\,2\right)$ = $9$$$