(See Desc.) A manufacturer can produce 4860 cell phones when x dollars is spent on labor and y dollars is spent on capital. The equation that relates x and y is (90x^3/4)(y^1/4)=4860.

Question

a. Find a formula in terms of x and y for dy/dx.

b. Find the value of dy/dx at the point (81,16).

Answer 1

You can either use implicit differentiation directly or take the logs first. As4860/90 = 54

3/4 ln x + 1/4 ln y = ln 54

3 ln x + ln y = 4 ln 54

Taking derivatives,

3/x dx + 1/y dy = 0

dy/dx = -3y/x

At (81,16) dy/dx = -3(16)/81 = -16/27

Note that , from 3 ln x + ln y = 4 ln 54, we could have taken the derivative without using implicit differentiation.

y = 54^4/x^3 = 3^12*2^4/x^3

Then, dy/dx = -3^13*2^4/x^4

Plugging in x = 81 = 3^4, we get -3^13*2^4/3^16 = -2^4/3^3 = -16/27