Chapter 14.3, Problem 12E

### 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 joint cost function for two products is C ( x , y ) = y  ln ( x + 1 ) dollars Find the marginal cost with respect to (a) x. (b) y.

(a)

To determine

To calculate: The marginal cost with respect to x if the joint cost function for two products is C(x,y)=yln(x+1) dollars.

Explanation

Given Information:

The joint cost function for the two products is C(x,y)=yln(x+1) dollars.

Formula used:

For a joint cost function, of the form C(x,y), the marginal cost with respect to x is given by âˆ‚Câˆ‚x and the marginal cost with respect to y is given by âˆ‚Câˆ‚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.

A derivative of natural logarithmic functions,

If y=lnu, where u is a differentiable function of x then dydx=1uâ‹…dudx.

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

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

(b)

To determine

To calculate: The marginal cost with respect to y if the joint cost function for two products is C(x,y)=yln(x+1) dollars.

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