Partial

at $\left(x\,y\right)$ 

at $\left(a\,b\right)$ 

$f_{\mathrm{xx}}$

$-y^2\sin \left(xy\right)$

$-\frac{1}{9}{\mathrm{\pi }}^2\sin \left(\frac{1}{18}{\mathrm{\pi }}^2\right)$

$f_{\mathrm{xy}}$

$\cos \left(xy\right)-yx\sin \left(xy\right)$

$\cos \left(\frac{1}{18}{\mathrm{\pi }}^2\right)-\frac{1}{18}{\mathrm{\pi }}^2\sin \left(\frac{1}{18}{\mathrm{\pi }}^2\right)$

$f_{\mathrm{yx}}$

$\cos \left(xy\right)-yx\sin \left(xy\right)$

$\cos \left(\frac{1}{18}{\mathrm{\pi }}^2\right)-\frac{1}{18}{\mathrm{\pi }}^2\sin \left(\frac{1}{18}{\mathrm{\pi }}^2\right)$

$f_{\mathrm{yy}}$

$-x^2\sin \left(xy\right)$

$-\frac{1}{36}{\mathrm{\pi }}^2\sin \left(\frac{1}{18}{\mathrm{\pi }}^2\right)$

Table 4.2.3(a)   Second partials of $f$ at $\left(x\,y\right)$ and $\left(\mathrm{\π}\/6\,\mathrm{\π}\/3\right)$ 

$f\=\sin \left(x y\right)$$\stackrel{\text{assign}}{\to }$

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

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

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

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

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

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)$$$

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

$\mathrm{f\_\_xx}\left(x\,y\right)$ = $-y^2\sin \left(xy\right)$$$$$

$\mathrm{f\_\_xx}\left(\frac{\mathrm{\pi }}{6}\,\frac{\mathrm{\pi }}{3}\right)$ = $-\frac{1}{9}{\mathrm{\π}}^2\sin \left(\frac{1}{18}{\mathrm{\π}}^2\right)$$$

$\mathrm{f\_\_xy}\left(x\,y\right)$ = $\cos \left(xy\right)-yx\sin \left(xy\right)$$$$$

$\mathrm{f\_\_xy}\left(\frac{\mathrm{\pi }}{6}\,\frac{\mathrm{\pi }}{3}\right)$ = $\cos \left(\frac{1}{18}{\mathrm{\π}}^2\right)-\frac{1}{18}{\mathrm{\π}}^2\sin \left(\frac{1}{18}{\mathrm{\π}}^2\right)$$$

$\mathrm{f\_\_yx}\left(x\,y\right)$ = $\cos \left(xy\right)-yx\sin \left(xy\right)$$$$$

$\mathrm{f\_\_yx}\left(\frac{\mathrm{\pi }}{6}\,\frac{\mathrm{\pi }}{3}\right)$ = $\cos \left(\frac{1}{18}{\mathrm{\π}}^2\right)-\frac{1}{18}{\mathrm{\π}}^2\sin \left(\frac{1}{18}{\mathrm{\π}}^2\right)$$$

$\mathrm{f\_\_yy}\left(x\,y\right)$ = $-x^2\sin \left(xy\right)$$$$$

$\mathrm{f\_\_yy}\left(\frac{\mathrm{\pi }}{6}\,\frac{\mathrm{\pi }}{3}\right)$ = $-\frac{1}{36}{\mathrm{\π}}^2\sin \left(\frac{1}{18}{\mathrm{\π}}^2\right)$$$

$f≔\sin \left(x y\right)\:$

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

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

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

$-y^2\sin \left(xy\right)$

$\mathrm{eval}\left(\mathrm{f\_\_xx}\,\left[x\=\mathrm{\π}\/6\,y\=\mathrm{\π}\/3\right]\right)$

$-\frac{1}{9}{\mathrm{\π}}^2\sin \left(\frac{1}{18}{\mathrm{\π}}^2\right)$

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

$\cos \left(xy\right)-yx\sin \left(xy\right)$

$\mathrm{eval}\left(\mathrm{f\_\_xy}\,\left[x\=\mathrm{\π}\/6\,y\=\mathrm{\π}\/3\right]\right)$

$\cos \left(\frac{1}{18}{\mathrm{\π}}^2\right)-\frac{1}{18}{\mathrm{\π}}^2\sin \left(\frac{1}{18}{\mathrm{\π}}^2\right)$

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

$\cos \left(xy\right)-yx\sin \left(xy\right)$

$\mathrm{eval}\left(\mathrm{f\_\_yx}\,\left[x\=\mathrm{\π}\/6\,y\=\mathrm{\π}\/3\right]\right)$

$\cos \left(\frac{1}{18}{\mathrm{\π}}^2\right)-\frac{1}{18}{\mathrm{\π}}^2\sin \left(\frac{1}{18}{\mathrm{\π}}^2\right)$

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

$-x^2\sin \left(xy\right)$

$\mathrm{eval}\left(\mathrm{f\_\_yy}\,\left[x\=\mathrm{\π}\/6\,y\=\mathrm{\π}\/3\right]\right)$

$-\frac{1}{36}{\mathrm{\π}}^2\sin \left(\frac{1}{18}{\mathrm{\π}}^2\right)$

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

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

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

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

$-y^2\sin \left(xy\right)$

${\mathrm{D}}_{1\,1}\left(f\right)\left(\mathrm{\π}\/6\,\mathrm{\π}\/3\right)$

$-\frac{1}{9}{\mathrm{\π}}^2\sin \left(\frac{1}{18}{\mathrm{\π}}^2\right)$

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

$\cos \left(xy\right)-yx\sin \left(xy\right)$

${\mathrm{D}}_{1\,2}\left(f\right)\left(\mathrm{\π}\/6\,\mathrm{\π}\/3\right)$

$\cos \left(\frac{1}{18}{\mathrm{\π}}^2\right)-\frac{1}{18}{\mathrm{\π}}^2\sin \left(\frac{1}{18}{\mathrm{\π}}^2\right)$

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

$\cos \left(xy\right)-yx\sin \left(xy\right)$

${\mathrm{D}}_{2\,1}\left(f\right)\left(\mathrm{\π}\/6\,\mathrm{\π}\/3\right)$

$\cos \left(\frac{1}{18}{\mathrm{\π}}^2\right)-\frac{1}{18}{\mathrm{\π}}^2\sin \left(\frac{1}{18}{\mathrm{\π}}^2\right)$

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

$-x^2\sin \left(xy\right)$

${\mathrm{D}}_{2\,2}\left(f\right)\left(\mathrm{\π}\/6\,\mathrm{\π}\/3\right)$

$-\frac{1}{36}{\mathrm{\π}}^2\sin \left(\frac{1}{18}{\mathrm{\π}}^2\right)$