Chapter 13 - Partial Derivatives - 13.3 Partial Derivatives - Exercises Set 13.3 - Page 937: 43

$$\eqalign{ & {f_x}\left( {x,y,z} \right) = \frac{{2z}}{x} \cr & {f_y}\left( {x,y,z} \right) = \frac{z}{y} \cr & {f_y}\left( {x,y,z} \right) = \ln \left( {{x^2}y\cos z} \right) - z\tan z \cr} $$

$$\eqalign{ & f\left( {x,y,z} \right) = z\ln \left( {{x^2}y\cos z} \right) \cr & {\text{Calculate }}{f_x}\left( {x,y,z} \right) \cr & {f_x}\left( {x,y,z} \right) = \frac{\partial }{{\partial x}}\left[ {z\ln \left( {{x^2}y\cos z} \right)} \right] \cr & {f_x}\left( {x,y,z} \right) = z\frac{\partial }{{\partial x}}\left[ {\ln \left( {{x^2}y\cos z} \right)} \right] \cr & {f_x}\left( {x,y,z} \right) = z\left( {\frac{{\frac{\partial }{{\partial x}}\left[ {{x^2}y\cos z} \right]}}{{{x^2}y\cos z}}} \right) \cr & {f_x}\left( {x,y,z} \right) = z\left( {\frac{{2xy\cos z}}{{{x^2}y\cos z}}} \right) \cr & {f_x}\left( {x,y,z} \right) = z\left( {\frac{2}{x}} \right) \cr & {f_x}\left( {x,y,z} \right) = \frac{{2z}}{x} \cr & \cr & {\text{Calculate }}{f_y}\left( {x,y,z} \right) \cr & {f_y}\left( {x,y,z} \right) = \frac{\partial }{{\partial y}}\left[ {z\ln \left( {{x^2}y\cos z} \right)} \right] \cr & {f_y}\left( {x,y,z} \right) = z\frac{\partial }{{\partial y}}\left[ {\ln \left( {{x^2}y\cos z} \right)} \right] \cr & {f_y}\left( {x,y,z} \right) = z\left[ {\frac{{{x^2}\cos z}}{{{x^2}y\cos z}}} \right] \cr & {f_y}\left( {x,y,z} \right) = z\left[ {\frac{1}{y}} \right] \cr & {f_y}\left( {x,y,z} \right) = \frac{z}{y} \cr & \cr & {\text{Calculate }}{f_z}\left( {x,y,z} \right) \cr & {f_z}\left( {x,y,z} \right) = \frac{\partial }{{\partial z}}\left[ {z\ln \left( {{x^2}y\cos z} \right)} \right] \cr & {\text{Use product rule}} \cr & {f_z}\left( {x,y,z} \right) = z\frac{\partial }{{\partial z}}\left[ {\ln \left( {{x^2}y\cos z} \right)} \right] + \ln \left( {{x^2}y\cos z} \right)\frac{\partial }{{\partial z}}\left[ z \right] \cr & {f_z}\left( {x,y,z} \right) = z\left( {\frac{{ - {x^2}y\sin z}}{{{x^2}y\cos z}}} \right) + \ln \left( {{x^2}y\cos z} \right)\left( 1 \right) \cr & {f_z}\left( {x,y,z} \right) = z\left( { - \tan z} \right) + \ln \left( {{x^2}y\cos z} \right) \cr & {f_y}\left( {x,y,z} \right) = \ln \left( {{x^2}y\cos z} \right) - z\tan z \cr} $$