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Prove the marginal rate of substitution more generally, that is given(a) $L\left(x_{1}, x_{2}, \ldots, x_{n}\right)=f\left(x_{1}, x_{2}, \ldots, x_{n}\right)+\lambda\left(a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}\right)$then $$\frac{a_{i}}{a_{j}}=\frac{\frac{\partial f}{\partial x_{i}}}{\frac{\partial f}{\partial x_{j}}}$$at the optimal point.(b) $L\left(x_{1}, x_{2}, \ldots, x_{n}\right)=a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}+\lambda f\left(x_{1}, x_{2}, \ldots, x_{n}\right)$then$$\frac{a_{i}}{a_{j}}=\frac{\frac{\partial f}{\partial x_{i}}}{\frac{\partial f}{\partial x_{j}}}$$at the optimal point.

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 5

Economic Applications

Partial Derivatives

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University of Michigan - Ann Arbor

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Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

04:14

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