Chapter 12 - Functions of Several Veriables - 12.4 Partial Derivatives - 12.4 Exercises - Page 904: 53

$$\eqalign{ & {h_w}\left( {w,x,y,z} \right) = \frac{z}{{xy}} \cr & {h_x}\left( {w,x,y,z} \right) = - \frac{{wz}}{{{x^2}y}} \cr & {h_y}\left( {w,x,y,z} \right) = - \frac{{wz}}{{x{y^2}}} \cr & {h_z}\left( {w,x,y,z} \right) = \frac{w}{{xy}} \cr} $$

$$\eqalign{ & h\left( {w,x,y,z} \right) = \frac{{wz}}{{xy}} \cr & {\text{Find the first partial derivative }}{h_w}\left( {w,x,y,z} \right) \cr & {h_w}\left( {w,x,y,z} \right) = \frac{\partial }{{\partial w}}\left[ {\frac{{wz}}{{xy}}} \right] \cr & {\text{treat }}x,y{\text{ and }}z{\text{ as a constants}} \cr & {h_w}\left( {w,x,y,z} \right) = \frac{z}{{xy}}\frac{\partial }{{\partial w}}\left[ w \right] \cr & {h_w}\left( {w,x,y,z} \right) = \frac{z}{{xy}}\left( 1 \right) \cr & {h_w}\left( {w,x,y,z} \right) = \frac{z}{{xy}} \cr & \cr & {\text{Find the first partial derivative }}{h_x}\left( {w,x,y,z} \right) \cr & {h_x}\left( {w,x,y,z} \right) = \frac{\partial }{{\partial w}}\left[ {\frac{{wz}}{{xy}}} \right] \cr & {\text{treat }}w,y{\text{ and }}z{\text{ as a constants}} \cr & {h_x}\left( {w,x,y,z} \right) = \frac{{wz}}{y}\frac{\partial }{{\partial w}}\left[ {\frac{1}{x}} \right] \cr & {h_x}\left( {w,x,y,z} \right) = \frac{{wz}}{y}\left( { - \frac{1}{{{x^2}}}} \right) \cr & {h_x}\left( {w,x,y,z} \right) = - \frac{{wz}}{{{x^2}y}} \cr & \cr & {\text{Find the first partial derivative }}{h_y}\left( {w,x,y,z} \right) \cr & {h_y}\left( {w,x,y,z} \right) = \frac{\partial }{{\partial y}}\left[ {\frac{{wz}}{{xy}}} \right] \cr & {\text{treat }}w,x{\text{ and }}z{\text{ as a constants}} \cr & {h_y}\left( {w,x,y,z} \right) = \frac{{wz}}{x}\frac{\partial }{{\partial y}}\left[ {\frac{1}{y}} \right] \cr & {h_y}\left( {w,x,y,z} \right) = \frac{{wz}}{x}\left( { - \frac{1}{{{y^2}}}} \right) \cr & {h_y}\left( {w,x,y,z} \right) = - \frac{{wz}}{{x{y^2}}} \cr & \cr & {\text{Find the first partial derivative }}{h_z}\left( {w,x,y,z} \right) \cr & {h_z}\left( {w,x,y,z} \right) = \frac{\partial }{{\partial z}}\left[ {\frac{{wz}}{{xy}}} \right] \cr & {\text{treat }}w,x{\text{ and }}z{\text{ as a constants}} \cr & {h_z}\left( {w,x,y,z} \right) = \frac{w}{{xy}}\frac{\partial }{{\partial z}}\left[ z \right] \cr & {h_z}\left( {w,x,y,z} \right) = \frac{w}{{xy}}\left( 1 \right) \cr & {h_z}\left( {w,x,y,z} \right) = \frac{w}{{xy}} \cr} $$