Chapter 13 - Partial Derivatives - 13.3 Partial Derivatives - Exercises Set 13.3 - Page 936: 2

$$\eqalign{ & {\bf{a}})\,\,\,\frac{{\partial z}}{{\partial x}} = 2{e^{2x}}\sin y,\,\,\,\,{\bf{b}})\,\,\,\,\frac{{\partial z}}{{\partial y}} = {e^{2x}}\cos y,\,\,{\bf{c}})\,\,\,2\sin y \cr & {\bf{d}})\,\,\,\,0,\,\,\,{\bf{e}}){f_y}\left( {1,y} \right) = \cos y,\,\,{\bf{f}})\,\,{f_y}\left( {1,y} \right) = {e^{2x}},\, \cr & {\bf{g}})\,\,\,\,0\,\,\,\,{\bf{h}})\,\,{f_y}\left( {1,2} \right) = 4 \cr} $$

$$\eqalign{ & {\text{Let }}z = {e^{2x}}\sin y \cr & {\bf{a}})\,\,{\text{Calculate the partial derivative }}\partial z/\partial x,{\text{ consider }}y{\text{ as a constant}} \cr & \,\,\,\,\frac{{\partial z}}{{\partial x}} = \frac{\partial }{{\partial x}}\left[ {{e^{2x}}\sin y} \right] \cr & \,\,\,\,\frac{{\partial z}}{{\partial x}} = 2{e^{2x}}\sin y \cr & \cr & {\bf{b}})\,\,{\text{Calculate the partial derivative }}{f_y}\left( {x,y} \right),{\text{ consider }}x{\text{ as a constant}} \cr & \,\,\,\,\frac{{\partial z}}{{\partial y}} = \frac{\partial }{{\partial y}}\left[ {{e^{2x}}\sin y} \right] \cr & \,\,\,\,\frac{{\partial z}}{{\partial y}} = {e^{2x}}\left( {\cos y} \right) \cr & \,\,\,\,\frac{{\partial z}}{{\partial y}} = {e^{2x}}\cos y \cr & \cr & {\bf{c}})\,\,{\text{Find the partial derivative }}{f_x}\left( {0,y} \right),{\text{ }} \cr & \,\,\,\,{\left[ {\frac{{\partial z}}{{\partial x}}} \right]_{\left( {0,y} \right)}} = 2{e^{2\left( 0 \right)}}\sin y \cr & \,\,\,\,\,{\left[ {\frac{{\partial z}}{{\partial x}}} \right]_{\left( {0,y} \right)}} = 2\sin y \cr & \cr & {\bf{d}})\,\,{\text{Find the partial derivative }}{f_x}\left( {x,0} \right),{\text{ }} \cr & \,\,\,\,{\left[ {\frac{{\partial z}}{{\partial x}}} \right]_{\left( {x,0} \right)}} = 2{e^{2\left( x \right)}}\sin \left( 0 \right) \cr & \,\,\,\,{\left[ {\frac{{\partial z}}{{\partial x}}} \right]_{\left( {x,0} \right)}} = 0 \cr & \cr & {\bf{e}})\,\,{\text{Find the partial derivative }}{f_y}\left( {0,y} \right),{\text{ }} \cr & \,\,{\left[ {\frac{{\partial z}}{{\partial y}}} \right]_{\left( {0,y} \right)}} = {e^{2\left( 0 \right)}}\cos y \cr & \,\,{\left[ {\frac{{\partial z}}{{\partial y}}} \right]_{\left( {0,y} \right)}} = \cos y \cr & \cr & {\bf{f}})\,\,{\text{Find the partial derivative }}{f_y}\left( {x,0} \right),{\text{ }} \cr & \,\,{\left[ {\frac{{\partial z}}{{\partial y}}} \right]_{\left( {x,0} \right)}} = {e^{2\left( x \right)}}\cos \left( 0 \right) \cr & \,\,{\left[ {\frac{{\partial z}}{{\partial y}}} \right]_{\left( {x,0} \right)}} = {e^{2x}} \cr & \cr & {\bf{g}})\,\,{\text{Find the partial derivative }}{f_x}\left( {\ln 2,0} \right),{\text{ }} \cr & \,\,{\left[ {\frac{{\partial z}}{{\partial x}}} \right]_{\left( {\ln 2,0} \right)}} = 2{e^{2\left( {\ln 2} \right)}}\sin \left( 0 \right) \cr & \,\,{\left[ {\frac{{\partial z}}{{\partial x}}} \right]_{\left( {\ln 2,0} \right)}} = 2{e^{\left( {\ln 4} \right)}}\left( 0 \right) \cr & \,\,{\left[ {\frac{{\partial z}}{{\partial x}}} \right]_{\left( {\ln 2,0} \right)}} = 0 \cr & \cr & {\bf{h}})\,\,{\text{Find the partial derivative }}{f_y}\left( {\ln 2,0} \right),{\text{ }} \cr & \,\,{\left[ {\frac{{\partial z}}{{\partial y}}} \right]_{\left( {\ln 2,0} \right)}} = {e^{2\left( {\ln 2} \right)}}\cos \left( 0 \right) \cr & \,\,{\left[ {\frac{{\partial z}}{{\partial y}}} \right]_{\left( {\ln 2,0} \right)}} = {e^{\left( {\ln 4} \right)}}\left( 1 \right) \cr & \,\,{\left[ {\frac{{\partial z}}{{\partial y}}} \right]_{\left( {\ln 2,0} \right)}} = 4 \cr} $$