$\frac{\partial }{\partial x}\left(\sin \left(x y\right)\cos \left(x\/y\right)\right)\=y\cos \left(xy\right)\cos \left(\frac{x}{y}\right)-\frac{\sin \left(xy\right)\sin \left(\frac{x}{y}\right)}{y}$

$\frac{\partial }{\partial y}\left(\sin \left(x y\right)\cos \left(x\/y\right)\right)\=x\cos \left(xy\right)\cos \left(\frac{x}{y}\right)\+\frac{\sin \left(xy\right)x\sin \left(\frac{x}{y}\right)}{y^2}$

$\genfrac{}{}{0ex}{}{\frac{\partial f}{\partial x}}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{\left(x\,y\right)\=\left(1\,-1\right)}$ =  ${\sin }^2\left(1\right)-{\cos }^2\left(1\right)$

$\genfrac{}{}{0ex}{}{\frac{\partial f}{\partial y}}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{\left(x\,y\right)\=\left(1\,-1\right)}$=  $1$

Table 4.1.3(a)   First partial derivatives of $f$

Context Panel

$f_x$ 

$f_y$ 

$\sin \left(x y\right)\cos \left(x\/y\right)$

$\sin \left(xy\right)\cos \left(\frac{x}{y}\right)$

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

$y\cos \left(xy\right)\cos \left(\frac{x}{y}\right)-\frac{\sin \left(xy\right)\sin \left(\frac{x}{y}\right)}{y}$

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

$-{\cos \left(1\right)}^2\+{\sin \left(1\right)}^2$

$\sin \left(x y\right)\cos \left(x\/y\right)$

$\sin \left(xy\right)\cos \left(\frac{x}{y}\right)$

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

$x\cos \left(xy\right)\cos \left(\frac{x}{y}\right)\+\frac{\sin \left(xy\right)x\sin \left(\frac{x}{y}\right)}{y^2}$

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

${\cos \left(1\right)}^2\+{\sin \left(1\right)}^2$

$\stackrel{\text{simplify}}{\=}$

$1$

Define $f$ as an expression

$\sin \left(x y\right)\cos \left(x\/y\right)$$\stackrel{\text{assign to a name}}{\to }$$f$

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

$\frac{\partial }{\partial x} f$ = $y\cos \left(xy\right)\cos \left(\frac{x}{y}\right)-\frac{\sin \left(xy\right)\sin \left(\frac{x}{y}\right)}{y}$$$

$\frac{\partial }{\partial y} f$ = $x\cos \left(xy\right)\cos \left(\frac{x}{y}\right)\+\frac{\sin \left(xy\right)x\sin \left(\frac{x}{y}\right)}{y^2}$$$

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

$\genfrac{}{}{0ex}{}{\frac{\partial }{\partial x} f}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=1\,y\=-1}$ = $-{\cos \left(1\right)}^2\+{\sin \left(1\right)}^2$$$

$\genfrac{}{}{0ex}{}{\frac{\partial }{\partial y} f}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=1\,y\=-1}$ = ${\cos \left(1\right)}^2\+{\sin \left(1\right)}^2$$\stackrel{\text{simplify}}{\=}$$1$$$

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\cos \left(xy\right)\cos \left(\frac{x}{y}\right)-\frac{\sin \left(xy\right)\sin \left(\frac{x}{y}\right)}{y}$$$

$\mathrm{f\_\_y}\left(x\,y\right)$ = $x\cos \left(xy\right)\cos \left(\frac{x}{y}\right)\+\frac{\sin \left(xy\right)x\sin \left(\frac{x}{y}\right)}{y^2}$$$

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

$\mathrm{f\_\_x}\left(1\,-1\right)$ = $-{\cos \left(1\right)}^2\+{\sin \left(1\right)}^2$$$

$\mathrm{f\_\_y}\left(1\,-1\right)$ = ${\cos \left(1\right)}^2\+{\sin \left(1\right)}^2$$\stackrel{\text{simplify}}{\=}$$1$$$

Define $f$ as an expression

$f≔\sin \left(x y\right)\cos \left(x\/y\right)\:$

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

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

$y\cos \left(xy\right)\cos \left(\frac{x}{y}\right)-\frac{\sin \left(xy\right)\sin \left(\frac{x}{y}\right)}{y}$

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

$x\cos \left(xy\right)\cos \left(\frac{x}{y}\right)\+\frac{\sin \left(xy\right)x\sin \left(\frac{x}{y}\right)}{y^2}$

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

$\mathrm{eval}\left(\mathrm{fx}\,\left[x\=1\,y\=-1\right]\right)$

$-{\cos \left(1\right)}^2\+{\sin \left(1\right)}^2$

$\mathrm{simplify}\left(\mathrm{eval}\left(\mathrm{fy}\,\left[x\=1\,y\=-1\right]\right)\right)$

$1$

Define $f$ as a function

$f≔\left(x\,y\right)\to \sin \left(x y\right)\cos \left(x\/y\right)\:$

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\cos \left(xy\right)\cos \left(\frac{x}{y}\right)-\frac{\sin \left(xy\right)\sin \left(\frac{x}{y}\right)}{y}$$$

${\mathrm{D}}_2\left(f\right)\left(x\,y\right)$ = $x\cos \left(xy\right)\cos \left(\frac{x}{y}\right)\+\frac{\sin \left(xy\right)x\sin \left(\frac{x}{y}\right)}{y^2}$$$

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

${\mathrm{D}}_1\left(f\right)\left(1\,-1\right)$ = $-{\cos \left(1\right)}^2\+{\sin \left(1\right)}^2$$$

$\mathrm{simplify}\left({\mathrm{D}}_2\left(f\right)\left(1\,-1\right)\right)$ = $1$$$