Chapter 13 - Partial Derivatives - 13.3 Partial Derivatives - Exercises Set 13.3 - Page 936: 8

$$\frac{\partial }{{\partial x}}\left[ {x{e^{\sqrt {15xy} }}} \right] = \left( {1 + \frac{{\sqrt {15xy} }}{2}} \right){e^{\sqrt {15xy} }}{\text{ and }}\frac{\partial }{{\partial y}}\left[ {x{e^{\sqrt {15xy} }}} \right] = \frac{{x{e^{\sqrt {15xy} }}\sqrt {15xy} }}{{2y}}$$

$$\eqalign{ & {\text{Calculate }}\frac{\partial }{{\partial x}}\left[ {x{e^{\sqrt {15xy} }}} \right] \cr & {\text{Differentiating with respect to }}x,{\text{ treat }}y{\text{ as a constant.}} \cr & {\text{Use the product rule}} \cr & \frac{\partial }{{\partial x}}\left[ {x{e^{\sqrt {15xy} }}} \right] = \frac{\partial }{{\partial x}}\left[ x \right]{e^{\sqrt {15xy} }} + \frac{\partial }{{\partial x}}\left[ {{e^{\sqrt {15xy} }}} \right]x \cr & \frac{\partial }{{\partial x}}\left[ {x{e^{\sqrt {15xy} }}} \right] = \left( 1 \right){e^{\sqrt {15xy} }} + \frac{\partial }{{\partial x}}\left[ {{e^{\sqrt {15xy} }}} \right]x \cr & \frac{\partial }{{\partial x}}\left[ {x{e^{\sqrt {15xy} }}} \right] = {e^{\sqrt {15xy} }} + x{e^{\sqrt {15xy} }}\left( {\frac{{15y}}{{2\sqrt {15xy} }}} \right) \cr & {\text{simplifying}} \cr & \frac{\partial }{{\partial x}}\left[ {x{e^{\sqrt {15xy} }}} \right] = {e^{\sqrt {15xy} }} + \frac{{15xy{e^{\sqrt {15xy} }}}}{{2\sqrt {15xy} }} \cr & \frac{\partial }{{\partial x}}\left[ {x{e^{\sqrt {15xy} }}} \right] = {e^{\sqrt {15xy} }} + \frac{{\sqrt {15xy} {e^{\sqrt {15xy} }}}}{2} \cr & \frac{\partial }{{\partial x}}\left[ {x{e^{\sqrt {15xy} }}} \right] = \left( {1 + \frac{{\sqrt {15xy} }}{2}} \right){e^{\sqrt {15xy} }} \cr & \cr & \frac{\partial }{{\partial y}}\left[ {x{e^{\sqrt {15xy} }}} \right] \cr & {\text{Differentiating with respect to }}y,{\text{ treat }}x{\text{ as a constant}} \cr & \frac{\partial }{{\partial y}}\left[ {x{e^{\sqrt {15xy} }}} \right] = x\frac{\partial }{{\partial y}}\left[ {{e^{\sqrt {15xy} }}} \right] \cr & \frac{\partial }{{\partial y}}\left[ {x{e^{\sqrt {15xy} }}} \right] = x{e^{\sqrt {15xy} }}\frac{\partial }{{\partial y}}\left[ {\sqrt {15xy} } \right] \cr & \frac{\partial }{{\partial y}}\left[ {x{e^{\sqrt {15xy} }}} \right] = x{e^{\sqrt {15xy} }}\left( {\frac{{15x}}{{2\sqrt {15xy} }}} \right) \cr & {\text{simplifying}} \cr & \frac{\partial }{{\partial y}}\left[ {x{e^{\sqrt {15xy} }}} \right] = x{e^{\sqrt {15xy} }}\left( {\frac{{\sqrt {15xy} }}{{2y}}} \right) \cr & \frac{\partial }{{\partial y}}\left[ {x{e^{\sqrt {15xy} }}} \right] = \frac{{x{e^{\sqrt {15xy} }}\sqrt {15xy} }}{{2y}} \cr & \cr & {\text{Then}}{\text{, the partial derivatives are}} \cr & \frac{\partial }{{\partial x}}\left[ {x{e^{\sqrt {15xy} }}} \right] = \left( {1 + \frac{{\sqrt {15xy} }}{2}} \right){e^{\sqrt {15xy} }}{\text{ and }}\frac{\partial }{{\partial y}}\left[ {x{e^{\sqrt {15xy} }}} \right] = \frac{{x{e^{\sqrt {15xy} }}\sqrt {15xy} }}{{2y}} \cr} $$