The formula in terms of x and y for dy/dx is y/ -2x
The value of dy/dx at the point (1,1) is -0.5
The question is incomplete, the full question is given below:
200 cars can produced when x dollars is spent on labor and y dollars is spent on capital. the equation that relates x and y is 45 x^1/3 y^2/3= 200
a. Find a formula in terms of x and y for dy/dx
b. Find the value of dy/dx at the point (1,1). Round your answer to 4 decimal places
Part a
A formula in terms of x and y for dy/dx
We differentiate equation that relates x and y with respect to x by implicit differentiation:
=d/dx(45 x^1/3 y^2/3) =d/dx( 200)
=> 45 *(1/3) x^-2/3 y^2/3 + 45 x^1/3 2/3 y^ -1/3 dy/dx=0
=> x^-2/3y^2/3+ 2 x^1/3 y^-1/3 dy/dx=0
=> y/x= -2dy/dx
=> log y = -2 log x +log c
=> y= x^ -2 +c
putting x=y=1
c=0
then y= 1/x^2
b) y/x= -2dy/dx
put x=y=1 we get
-1/2 = dy/dx
dy/dx =-0.5
learn more about of differentiation here
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