Chapter 14.3, Problem 20E

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Chapter
Section

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# If a production function is given by z   =   12 3 / 4 y 1 / 3 , find the marginal productivity of (a) x. (b) y.

(a)

To determine

To calculate: The marginal productivity of x if a production function is given by z=12x3/4y1/3.

Explanation

Given Information:

The production function is given by z=12x3/4y1/3.

Formula used:

For a production function, of the form z=f(x,y), the marginal productivity of x is given by âˆ‚zâˆ‚x and the marginal productivity of y is given by âˆ‚zâˆ‚y.

For a function f(x,y), the partial derivative of f with respect to x is calculated by taking the derivative of f(x,y) with respect to x and keeping the other variable y constant. The partial derivative of f with respect to x is denoted by fx.

Power of x rule for a real number n is such that, if f(x)=xn then fâ€²(x)=nxnâˆ’1.

Constant function rule for a constant c is such that, if f(x)=c then fâ€²(x)=0.

The coefficient rule for a constant c is such that, if f(x)=câ‹…u(x), where u(x) is a differentiable function of x, then fâ€²(x)=câ‹…uâ€²(x).

Calculation:

The production function is z=12x3/4y1/3

(b)

To determine

To calculate: The marginal productivity of y if a production function is given by z=12x3/4y1/3.

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