Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 14 - Partial Derivatives - 14.3 Exercises - Page 938: 84

Answer

For $\displaystyle \frac{\partial P}{\partial L}$, L is the variable, all other variables are treated as constants. $\displaystyle \frac{\partial P}{\partial L}=\alpha bL^{\alpha-1}K^{\beta}$ For $\displaystyle \frac{\partial P}{\partial K}$, K is the variable, all other variables are treated as constants. $\displaystyle \frac{\partial P}{\partial K}=\beta bL^{\alpha}K^{\beta-1}$ LHS=$L\displaystyle \frac{\partial P}{\partial L}+K\frac{\partial P}{\partial K}$ $= L(\alpha bL^{\alpha-1}K^{\beta})+K(\beta bL^{\alpha}K^{\beta-1})$ $=\alpha bL^{\alpha}K^{\beta}+\beta bL^{\alpha}K^{\beta}$ $=(\alpha+\beta)bL^{\alpha}K^{\beta}$ $=(\alpha+\beta)P$ =RHS

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