Chapter 12 - Functions of Several Veriables - 12.4 Partial Derivatives - 12.4 Exercises - Page 904: 52

$$\eqalign{ & {g_w}\left( {w,x,y,z} \right) = - \sin \left( {y - z} \right)sin\left( {w + x} \right) \cr & {g_x}\left( {w,x,y,z} \right) = - sin\left( {w + x} \right)\sin \left( {y - z} \right) \cr & {g_y}\left( {w,x,y,z} \right) = \cos \left( {w + x} \right)\cos \left( {y - z} \right) \cr & {g_z}\left( {w,x,y,z} \right) = - \cos \left( {w + x} \right)cos\left( {y - z} \right) \cr} $$

$$\eqalign{ & g\left( {w,x,y,z} \right) = \cos \left( {w + x} \right)\sin \left( {y - z} \right) \cr & {\text{Find the first partial derivative }}{g_w}\left( {w,x,y,z} \right) \cr & {g_w}\left( {w,x,y,z} \right) = \frac{\partial }{{\partial w}}\left[ {\cos \left( {w + x} \right)\sin \left( {y - z} \right)} \right] \cr & {\text{treat }}x,y{\text{ and }}z{\text{ as a constants}} \cr & {g_w}\left( {w,x,y,z} \right) = \sin \left( {y - z} \right)\frac{\partial }{{\partial w}}\left[ {\cos \left( {w + x} \right)} \right] \cr & {g_w}\left( {w,x,y,z} \right) = \sin \left( {y - z} \right)\left[ { - sin\left( {w + x} \right)} \right] \cr & {g_w}\left( {w,x,y,z} \right) = - \sin \left( {y - z} \right)sin\left( {w + x} \right) \cr & \cr & {\text{Find the first partial derivative }}{f_x}\left( {w,x,y,z} \right) \cr & {g_x}\left( {w,x,y,z} \right) = \frac{\partial }{{\partial x}}\left[ {\cos \left( {w + x} \right)\sin \left( {y - z} \right)} \right] \cr & {\text{treat }}w,y{\text{ and }}z{\text{ as a constants}} \cr & {g_x}\left( {w,x,y,z} \right) = \sin \left( {y - z} \right)\frac{\partial }{{\partial x}}\left[ {\cos \left( {w + x} \right)} \right] \cr & {g_x}\left( {w,x,y,z} \right) = \sin \left( {y - z} \right)\left[ { - sin\left( {w + x} \right)} \right] \cr & {g_x}\left( {w,x,y,z} \right) = - sin\left( {w + x} \right)\sin \left( {y - z} \right) \cr & \cr & {\text{Find the first partial derivative }}{f_y}\left( {w,x,y,z} \right) \cr & {g_y}\left( {w,x,y,z} \right) = \frac{\partial }{{\partial y}}\left[ {\cos \left( {w + x} \right)\sin \left( {y - z} \right)} \right] \cr & {\text{treat }}w,x{\text{ and }}z{\text{ as a constants}} \cr & {g_y}\left( {w,x,y,z} \right) = \cos \left( {w + x} \right)\frac{\partial }{{\partial y}}\left[ {\sin \left( {y - z} \right)} \right] \cr & {g_y}\left( {w,x,y,z} \right) = \cos \left( {w + x} \right)\left[ {\cos \left( {y - z} \right)} \right] \cr & {g_y}\left( {w,x,y,z} \right) = \cos \left( {w + x} \right)\cos \left( {y - z} \right) \cr & \cr & {\text{Find the first partial derivative }}{f_z}\left( {w,x,y,z} \right) \cr & {g_z}\left( {w,x,y,z} \right) = \frac{\partial }{{\partial z}}\left[ {\cos \left( {w + x} \right)\sin \left( {y - z} \right)} \right] \cr & {\text{treat }}w,x{\text{ and }}y{\text{ as a constants}} \cr & {g_z}\left( {w,x,y,z} \right) = \cos \left( {w + x} \right)\frac{\partial }{{\partial z}}\left[ {\sin \left( {y - z} \right)} \right] \cr & {g_z}\left( {w,x,y,z} \right) = \cos \left( {w + x} \right)\left[ {cos\left( {y - z} \right)} \right]\frac{\partial }{{\partial z}}\left[ {y - z} \right] \cr & {g_z}\left( {w,x,y,z} \right) = \cos \left( {w + x} \right)cos\left( {y - z} \right)\left( { - 1} \right) \cr & {g_z}\left( {w,x,y,z} \right) = - \cos \left( {w + x} \right)cos\left( {y - z} \right) \cr} $$