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: 61


$$ f_{yy}(2,3)=- \frac{4}{9}.$$

Work Step by Step

Since $ f(x,y)=x\ln ( y^2)$, then by using the chain rule, we have $$ f_y(x,y)=\frac{\partial f}{\partial y}=\frac{x}{y^2}(2y)=\frac{2x}{y}.$$ Again using the chain rule, we get $$ f_{yy}(x,y)=\frac{\partial^2 f}{\partial y^2}=- \frac{2x}{y^2}.$$ Hence, $$ f_{yy}(2,3)=- \frac{4}{9}.$$
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