Chapter 13 - Partial Derivatives - 13.5 The Chain Rule - Exercises Set 13.5 - Page 957: 33

$$\,{\left. {\frac{{\partial z}}{{\partial r}}} \right|_{r = 2,\theta = \pi /6}} = \frac{3}{2}{e^{\sqrt 3 }} + \frac{{\sqrt 3 }}{2}{e^{\sqrt 3 }}{\text{ and }}{\left. {\frac{{\partial z}}{{\partial \theta }}} \right|_{r = 2,\theta = \pi /6}} = \sqrt 3 {e^{\sqrt 3 }} + {e^{\sqrt 3 }}$$

$$\eqalign{ & {\text{Let }}z = xy{e^{x/y}},\,\,\,\,\,\,\,x = r\cos \theta ,\,\,\,\,\,\,\,\,y = r\sin \theta \cr & \,\,\,\,\,r = 2,\,\,\,\,\theta = \frac{\pi }{6},\,\,\,\,\,\,x = 2\cos \left( {\frac{\pi }{6}} \right) = \sqrt 3 \cr & \,\,\,\,\,r = 2,\,\,\,\,\theta = \frac{\pi }{6},\,\,\,\,\,\,y = 2\sin \left( {\frac{\pi }{6}} \right) = 1 \cr & {\text{Calculate the partial derivatives }}\frac{{\partial f}}{{\partial x}}{\text{ and }}\frac{{\partial f}}{{\partial y}} \cr & \,\,\,\,\,\,\frac{{\partial z}}{{\partial x}} = \frac{\partial }{{\partial x}}\left[ {xy{e^{x/y}}} \right] \cr & \,\,\,\,\,\,\frac{{\partial z}}{{\partial x}} = xy\left( {\frac{1}{y}{e^{x/y}}} \right) + y{e^{x/y}} \cr & \,\,\,\,\,\,\frac{{\partial z}}{{\partial x}} = x{e^{x/y}} + y{e^{x/y}} \cr & {\text{Calculate the partial derivatives }}\frac{{\partial x}}{{\partial u}}{\text{ and }}\frac{{\partial y}}{{\partial u}} \cr & \,\,\,\,\,\,\frac{{\partial z}}{{\partial y}} = \frac{\partial }{{\partial y}}\left[ {xy{e^{x/y}}} \right] \cr & \,\,\,\,\,\,\frac{{\partial z}}{{\partial y}} = xy\left( { - \frac{1}{{{y^2}}}{e^{x/y}}} \right) + x{e^{x/y}} \cr & \,\,\,\,\,\,\frac{{\partial z}}{{\partial y}} = - \frac{x}{y}{e^{x/y}} + x{e^{x/y}} \cr & {\text{Calculate the partial derivatives }}\frac{{\partial x}}{{\partial r}}{\text{ and }}\frac{{\partial y}}{{\partial r}} \cr & \,\,\,\,\,\,\frac{{\partial x}}{{\partial r}} = \frac{\partial }{{\partial r}}\left[ {r\cos \theta } \right] = \cos \theta \cr & \,\,\,\,\,\,\frac{{\partial y}}{{\partial r}} = \frac{\partial }{{\partial r}}\left[ {r\sin \theta } \right] = \sin \theta \cr & {\text{Calculate the partial derivatives }}\frac{{\partial x}}{{\partial \theta }}{\text{ and }}\frac{{\partial y}}{{\partial \theta }} \cr & \,\,\,\,\,\,\frac{{\partial x}}{{\partial \theta }} = \frac{\partial }{{\partial \theta }}\left[ {r\cos \theta } \right] = - r\sin \theta \cr & \,\,\,\,\,\,\frac{{\partial y}}{{\partial \theta }} = \frac{\partial }{{\partial \theta }}\left[ {r\sin \theta } \right] = r\cos \theta \cr & \cr & {\text{Calculate the partial derivative }}\frac{{\partial z}}{{\partial r}} \cr & \,\,\,\,\,\,\,\frac{{\partial z}}{{\partial r}} = \frac{{\partial z}}{{\partial x}}\frac{{\partial x}}{{\partial r}} + \frac{{\partial z}}{{\partial y}}\frac{{\partial y}}{{\partial r}} \cr & \,\,\,\,\,\,\,\frac{{\partial z}}{{\partial r}} = \left( {x{e^{x/y}} + y{e^{x/y}}} \right)\left( {\cos \theta } \right) + \left( { - \frac{x}{y}{e^{x/y}} + x{e^{x/y}}} \right)\sin \theta \cr & \,\,\,\,\,\,\,{\left. {\frac{{\partial z}}{{\partial r}}} \right|_{r = 2,\theta = \pi /6}} = \left( {\sqrt 3 {e^{\sqrt 3 /1}} + {e^{\sqrt 3 }}} \right)\left( {\cos \frac{\pi }{6}} \right) + \left( { - \frac{{\sqrt 3 }}{1}{e^{\sqrt 3 /1}} + \sqrt 3 {e^{\sqrt 3 /1}}} \right)\left( {\sin \frac{\pi }{6}} \right) \cr & \,\,\,\,\,\,\,{\left. {\frac{{\partial z}}{{\partial r}}} \right|_{r = 2,\theta = \pi /6}} = \frac{3}{2}{e^{\sqrt 3 }} + \frac{{\sqrt 3 }}{2}{e^{\sqrt 3 }} \cr & \cr & {\text{Calculate the partial derivative }}\frac{{\partial z}}{{\partial \theta }} \cr & \,\,\,\,\,\,\,\frac{{\partial z}}{{\partial \theta }} = \frac{{\partial z}}{{\partial x}}\frac{{\partial x}}{{\partial \theta }} + \frac{{\partial z}}{{\partial y}}\frac{{\partial y}}{{\partial \theta }} \cr & \,\,\,\,\,\,\,\frac{{\partial z}}{{\partial \theta }} = \left( {x{e^{x/y}} + y{e^{x/y}}} \right)\left( { - r\sin \theta } \right) + \left( { - \frac{x}{y}{e^{x/y}} + x{e^{x/y}}} \right)r\cos \theta \cr & \,\,\,\,\,\,\,{\left. {\frac{{\partial z}}{{\partial \theta }}} \right|_{r = 2,\theta = \pi /6}} = \left( {\sqrt 3 {e^{\sqrt 3 }} + {e^{\sqrt 3 }}} \right)\left( { - 2\sin \frac{\pi }{6}} \right) + \left( { - \sqrt 3 {e^{\sqrt 3 }} + \sqrt 3 {e^{\sqrt 3 }}} \right)\left( {2\cos \frac{\pi }{6}} \right) \cr & \,\,\,\,\,\,\,{\left. {\frac{{\partial z}}{{\partial \theta }}} \right|_{r = 2,\theta = \pi /6}} = \left( {\sqrt 3 {e^{\sqrt 3 }} + {e^{\sqrt 3 }}} \right)\left( { - 1} \right) + \left( 0 \right)\left( {2\sqrt 3 } \right) \cr & \,\,\,\,\,\,\,{\left. {\frac{{\partial z}}{{\partial \theta }}} \right|_{r = 2,\theta = \pi /6}} = \sqrt 3 {e^{\sqrt 3 }} + {e^{\sqrt 3 }} \cr} $$