Chapter 13 - Partial Derivatives - 13.3 Partial Derivatives - Exercises Set 13.3 - Page 937: 38

$${f_x}\left( {1,1} \right) = 3e,\,\,\,{\text{and }}\,\,\,{f_y}\left( {1,1} \right) = 2e$$

$$\eqalign{ & f\left( {x,y} \right) = {x^2}y{e^{xy}};\,\,\,\,\,\,\,\partial f/\partial x\left( {1,1} \right),\,\,\,\,\,\,\partial f/\partial y\left( {1,1} \right) \cr & \cr & {\text{Calculate }}{f_x}\left( {x,y} \right){\text{ differentiating with respect to }}x;{\text{ treat }}y{\text{ as a constant}} \cr & {\text{and use the product rule}} \cr & {f_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left( {{x^2}y{e^{xy}}} \right) \cr & {f_x}\left( {x,y} \right) = y{e^{xy}}\frac{\partial }{{\partial x}}\left( {{x^2}} \right) + {x^2}y\frac{\partial }{{\partial x}}\left( {{e^{xy}}} \right) \cr & {f_x}\left( {x,y} \right) = y{e^{xy}}\left( {2x} \right) + {x^2}y\left( {y{e^{xy}}} \right) \cr & {f_x}\left( {x,y} \right) = 2xy{e^{xy}} + {x^2}{y^2}{e^{xy}} \cr & \cr & {\text{Evaluating }}{f_x}\left( {1,1} \right) \cr & {f_x}\left( {1,1} \right) = 2\left( 1 \right)\left( 1 \right){e^{\left( 1 \right)\left( 1 \right)}} + {\left( 1 \right)^2}{\left( 1 \right)^2}{e^{\left( 1 \right)\left( 1 \right)}} \cr & {f_x}\left( {1,1} \right) = 2e + e \cr & {f_x}\left( {1,1} \right) = 3e \cr & \cr & {\text{Calculate }}{f_y}\left( {x,y} \right){\text{ differentiating with respect to }}y;{\text{ treat }}x{\text{ as a constant}} \cr & {\text{and use the product rule}} \cr & {f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left( {{x^2}y{e^{xy}}} \right) \cr & {f_y}\left( {x,y} \right) = {x^2}{e^{xy}}\frac{\partial }{{\partial y}}\left( y \right) + {x^2}y\frac{\partial }{{\partial y}}\left( {{e^{xy}}} \right) \cr & {f_y}\left( {x,y} \right) = y{e^{xy}}\left( 1 \right) + {x^2}y\left( {x{e^{xy}}} \right) \cr & {f_y}\left( {x,y} \right) = y{e^{xy}} + {x^3}y{e^{xy}} \cr & \cr & {\text{Evaluating }}{f_y}\left( {1,1} \right) \cr & {f_y}\left( {1,1} \right) = \left( 1 \right){e^{\left( 1 \right)\left( 1 \right)}} + {\left( 1 \right)^3}\left( 1 \right){e^{\left( 1 \right)\left( 1 \right)}} \cr & {f_y}\left( {1,1} \right) = e + e \cr & {f_y}\left( {1,1} \right) = 2e \cr} $$