Chapter 9 - Multiveriable Calculus - 9.2 Partial Derivatives - 9.2 Exercises - Page 478: 20

$$\eqalign{ & {f_x}\left( {x,y} \right) = 14x{e^{{x^2} + x + 2y}} + 8x{e^{{x^2}}} + 7{e^{{x^2} + x + 2y}} + 7{y^2}{e^{x + 2y}} + 14{e^{x + 2y}} \cr & {f_y}\left( {x,y} \right) = 14y{e^{x + 2y}} + 8y + 14{e^{{x^2} + x + 2y}} + 14{y^2}{e^{x + 2y}} + 28{e^{x + 2y}} \cr & {f_x}\left( {2, - 1} \right) = 51{e^4} + 21 \cr & {f_y}\left( { - 4,3} \right) = 196{e^2} + 14{e^{18}} + 24 \cr} $$

$$\eqalign{ & f\left( {x,y} \right) = \left( {7{e^{x + 2y}} + 4} \right)\left( {{e^{{x^2}}} + {y^2} + 2} \right) \cr & {\text{Find }}{f_x}\left( {x,y} \right){\text{ and }}{f_y}\left( {x,y} \right) \cr & \cr & {f_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {\left( {7{e^{x + 2y}} + 4} \right)\left( {{e^{{x^2}}} + {y^2} + 2} \right)} \right] \cr & {\text{treat y as a constant and }}x{\text{ as a variable and use the product rule}} \cr & {f_x}\left( {x,y} \right) = \left( {7{e^{x + 2y}} + 4} \right)\frac{\partial }{{\partial x}}\left[ {\left( {{e^{{x^2}}} + {y^2} + 2} \right)} \right] + \left( {{e^{{x^2}}} + {y^2} + 2} \right)\frac{\partial }{{\partial x}}\left[ {\left( {7{e^{x + 2y}} + 4} \right)} \right] \cr & {\text{solving derivatives}} \cr & {f_x}\left( {x,y} \right) = \left( {7{e^{x + 2y}} + 4} \right)\left( {2x{e^{{x^2}}}} \right) + \left( {{e^{{x^2}}} + {y^2} + 2} \right)\left( {7{e^{x + 2y}}} \right) \cr & {\text{multiplying}} \cr & {f_x}\left( {x,y} \right) = 14x{e^{{x^2} + x + 2y}} + 8x{e^{{x^2}}} + 7{e^{{x^2} + x + 2y}} + 7{y^2}{e^{x + 2y}} + 14{e^{x + 2y}} \cr & \cr & {\text{evaluate }}{f_x}\left( {2, - 1} \right) \cr & {f_x}\left( {2, - 1} \right) = 14\left( 2 \right){e^{{{\left( 2 \right)}^2} + \left( 2 \right) + 2\left( { - 1} \right)}} + 8\left( 2 \right){e^{{{\left( 2 \right)}^2}}} + 7{e^{{{\left( 2 \right)}^2} + \left( 2 \right) + 2\left( { - 1} \right)}} + 7{\left( { - 1} \right)^2}{e^{2 + 2\left( { - 1} \right)}} + 14{e^{2 + 2\left( { - 1} \right)}} \cr & {f_x}\left( {2, - 1} \right) = 28{e^4} + 16{e^4} + 7{e^4} + 7{e^0} + 14{e^0} \cr & {f_x}\left( {2, - 1} \right) = 51{e^4} + 21 \cr & \cr & {f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {\left( {7{e^{x + 2y}} + 4} \right)\left( {{e^{{x^2}}} + {y^2} + 2} \right)} \right] \cr & {\text{treat y as a constant and }}x{\text{ as a variable and use the product rule}} \cr & {f_y}\left( {x,y} \right) = \left( {7{e^{x + 2y}} + 4} \right)\frac{\partial }{{\partial y}}\left[ {\left( {{e^{{x^2}}} + {y^2} + 2} \right)} \right] + \left( {{e^{{x^2}}} + {y^2} + 2} \right)\frac{\partial }{{\partial y}}\left[ {\left( {7{e^{x + 2y}} + 4} \right)} \right] \cr & {\text{solving derivatives}} \cr & {f_y}\left( {x,y} \right) = \left( {7{e^{x + 2y}} + 4} \right)\left( {2y} \right) + \left( {{e^{{x^2}}} + {y^2} + 2} \right)\left( {14{e^{x + 2y}}} \right) \cr & {\text{multiplying}} \cr & {f_y}\left( {x,y} \right) = 14y{e^{x + 2y}} + 8y + 14{e^{{x^2} + x + 2y}} + 14{y^2}{e^{x + 2y}} + 28{e^{x + 2y}} \cr & \cr & {\text{evaluate }}{f_y}\left( { - 4,3} \right) \cr & {f_y}\left( { - 4,3} \right) = 14\left( 3 \right){e^{ - 4 + 2\left( 3 \right)}} + 8\left( 3 \right) + 14{e^{{{\left( { - 4} \right)}^2} + \left( { - 4} \right) + 2\left( 3 \right)}} + 14{\left( 3 \right)^2}{e^{ - 4 + 2\left( 3 \right)}} + 28{e^{ - 4 + 2\left( 3 \right)}} \cr & {f_y}\left( { - 4,3} \right) = 42{e^2} + 24 + 14{e^{18}} + 126{e^2} + 28{e^2} \cr & {f_y}\left( { - 4,3} \right) = 196{e^2} + 14{e^{18}} + 24 \cr} $$