Chapter 14.3, Problem 15E

### 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

# Suppose that the production function for a product is z = x  ln( y + 1 ) , where x represents the number of work-hours and y represents the available capital (per week). Find the marginal productivity of (a) x. (b) y.

(a)

To determine

To calculate: The marginal productivity of x if the production function for a product is z=xln(y+1), where x represents the number of work-hours and y represents the available capital (per week).

Explanation

Given Information:

The production function for a product is z=xln(y+1), where x represents the number of work-hours and y represents the available capital (per week).

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.

Chain rule for the function f(x)=u(v(x)) is fâ€²(x)=uâ€²(v(x))â‹…vâ€²(x).

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)

(b)

To determine

To calculate: The marginal productivity of y if the production function for a product is z=xln(y+1), where x represents the number of work-hours and y represents the available capital (per week).

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