Answered: Find the rate of change of y with…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Find the rate of change of y with respect to x when x=3000 and y=100

Expert Solution

Step 1

Given data:

The given expression of the function is 180x^0.75y^0.25=230,734.

The given value of units of labor is x=3000.

The given capital invested is y=100.

The differentiation of the given expression with respect to x on both sides is,

$\begin{array}{rcl} \frac{d}{d x} (180 x^{0.75} y^{0.25}) & = & \frac{d}{d x} (230, 734) \\ 180 x^{0.75} (0.25 y^{- 0.75} \frac{d y}{d x}) + 180 y^{0.25} (0.75 x^{- 0.25}) & = & 0 \\ 0.25 x^{0.75} y^{- 0.75} \frac{d y}{d x} + 0.75 x^{- 0.25} y^{0.25} & = & 0 \\ \frac{d y}{d x} & = & - 3 (\frac{y^{0.25 + 0.75}}{x^{0.75 + 0.25}}) \\ = & - \frac{3 y}{x} \end{array}$

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