Find the differential $dz$ of each function:

Solution (a) $f$ is defined everywhere in the $xy$-plane. The partial derivatives of $f$ are $$\begin{equation*} f_{x}(x,y) =e^{x}\cos\;y\qquad f_{y}(x,y) =-e^{x}\;\sin\;y \end{equation*}$$

Since $f_{x}$ and $f_{y}$ are continuous everywhere in the $xy$-plane, the function $z=f(x,y)$ is differentiable. The differential $dz$ is $$dz=e^{x}\cos\;y\,dx-e^{x}\sin\;y\,dy$$

(b) The domain of $f$ is $\left\{(x,y) \,|\,x>0,\,y\neq 0\right\} .$ The partial derivatives of $f$ are $$f_{x}(x,y) =\dfrac{1}{xy}\qquad f_{y}(x,y) =-\dfrac{\ln\;x}{y^{2}}$$

Since both partial derivatives exist and are continuous at every point $(x_{0},y_{0})$ in the domain of $f$, the function $z=f(x,y)$ is differentiable at every point $(x_{0},y_{0})$ in its domain. The differential $dz$ is $$dz=\dfrac{1}{xy}\, dx-\dfrac{\ln\;x}{y^{2}}\,dy$$