Chapter 13 - Partial Derivatives - 13.3 Partial Derivatives - Exercises Set 13.3 - Page 937: 42

$$\eqalign{ & {\text{Let }}w = {x^2}y\cos z \cr & {\text{a) Calculate the partial derivative }}\frac{{\partial w}}{{\partial x}} \cr & \frac{{\partial w}}{{\partial x}} = \frac{\partial }{{\partial y}}\left[ {{x^2}y\cos z} \right] \cr & \frac{{\partial w}}{{\partial x}} = y\cos z\frac{\partial }{{\partial y}}\left[ {{x^2}} \right] \cr & \frac{{\partial w}}{{\partial x}} = y\cos z\left( {2x} \right) \cr & \frac{{\partial w}}{{\partial x}} = 2xy\cos z \cr & \cr & {\text{b) Calculate the partial derivative }}\frac{{\partial w}}{{\partial y}} \cr & \frac{{\partial w}}{{\partial y}} = \frac{\partial }{{\partial x}}\left[ {{x^2}y\cos z} \right] \cr & \frac{{\partial w}}{{\partial y}} = {x^2}\cos z\frac{\partial }{{\partial x}}\left[ y \right] \cr & \frac{{\partial w}}{{\partial y}} = {x^2}\cos z\left( 1 \right) \cr & \frac{{\partial w}}{{\partial y}} = {x^2}\cos z \cr & \cr & {\text{c) Calculate the partial derivative }}\frac{{\partial w}}{{\partial z}} \cr & \frac{{\partial w}}{{\partial z}} = \frac{\partial }{{\partial z}}\left[ {{x^2}y\cos z} \right] \cr & \frac{{\partial w}}{{\partial z}} = {x^2}y\frac{\partial }{{\partial z}}\left[ {\cos z} \right] \cr & \frac{{\partial w}}{{\partial z}} = {x^2}y\left( { - \sin z} \right) \cr & \frac{{\partial w}}{{\partial z}} = - {x^2}y\sin z \cr & \cr & {\text{d) Calculate the partial derivative }}\frac{{\partial w}}{{\partial x}}{\text{ at the point }}\left( {2,y,z} \right) \cr & \frac{{\partial w}}{{\partial x}} = 2xy\cos z \cr & \partial w/\partial x\left( {2,y,z} \right) = 2\left( 2 \right)y\cos z \cr & \partial w/\partial x\left( {2,y,z} \right) = 4y\cos z \cr & \cr & {\text{e) Calculate the partial derivative }}\frac{{\partial w}}{{\partial y}}{\text{ at the point }}\left( {2,1,z} \right) \cr & \frac{{\partial w}}{{\partial y}} = {x^2}\cos z \cr & \partial w/\partial y\left( {2,1,z} \right) = {\left( 2 \right)^2}\cos z \cr & \partial w/\partial y\left( {2,1,z} \right) = 4\cos z \cr & \cr & {\text{f) Calculate the partial derivative }}\frac{{\partial w}}{{\partial z}}{\text{ at the point }}\left( {2,1,0} \right) \cr & \frac{{\partial w}}{{\partial z}} = - {x^2}y\sin z \cr & \partial w/\partial y\left( {2,1,0} \right) = - {\left( 2 \right)^2}\left( 1 \right)\sin 0 \cr & \partial w/\partial y\left( {2,1,0} \right) = - 4\left( 0 \right) \cr & \partial w/\partial y\left( {2,1,0} \right) = 0 \cr} $$

$$\eqalign{ & {\text{a)}}2xy\cos z,\,\,\,{\text{b)}}{x^2}\cos z,\,\,\,{\text{c)}} - {x^2}y\sin z \cr & {\text{d)}}4y\cos z,\,\,\,{\text{e)}}4\cos z,\,\,\,{\text{f)}}0 \cr} $$