For each function $z=f(x,y)$, find $f_{x}(x,y)$ and $f_{y}(x,y)$.

Solution  (a) To find $f_{x}(x,y)$, treat $y$ as a constant in $f(x,y)=3x^{2}y+2x-3y$ and differentiate with respect to $x$. The result is $$f_{x}(x,y)=6\;xy+2$$

To find $f_{y}(x,y)$, treat $x$ as a constant and differentiate with respect to $y$. The result is $$f_{y}(x,y)=3x^{2}-3$$

(b) For $f(x,y)=x\;\sin\;y+y\;\sin\;x$, we have