Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 15 - Differentiation in Several Variables - 15.3 Partial Derivatives - Exercises - Page 782: 62


$$ g_{xy}(-3,2)=24e^{6}.$$

Work Step by Step

Since $ g(x,y)=xe^{-xy}$, then by using the chain rule, we have $$ g_y(x,y)=\frac{\partial g}{\partial y}= -x^2e^{-xy}.$$ Using the product rule, we get $$ g_{xy}(x,y)=\frac{\partial^2 g}{\partial x\partial y}= -2xe^{-xy}+x^2ye^{-xy}=(-2+x)xe^{-xy}.$$ Plugging in the coordinates gives: $$ g_{xy}(-3,2)=24e^{6}.$$
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