The given equation is:
x y – 5000 = y
Rewrite this to create an explicit equation:
x y – y = 5000
y ( x – 1) = 5000
y = 5000 / (x – 1)
Derive to get dy / dx:
dy = - 5000 dx / (x – 1)^2
dy / dx = - 5000 / (x – 1)^2
So plugging in x = 6,
dy / dx = - 5000 / (6 – 1)^2
dy / dx = - 200
Answer:
[tex]\frac{dy}{dx}|_{x=6}= -200[/tex]
Step-by-step explanation:
Given : Demand equation [tex]xy − 5,000 = y[/tex]
To Find [tex]\frac{dy}{dx}|_{x=6}[/tex]
Solution :
[tex]xy- y= 5000[/tex]
[tex]y(x-1)= 5000[/tex]
[tex]y= \frac{5000}{x-1}[/tex]
Differentiating with respect to x
[tex]\frac{dy}{dx}= \frac{-5000}{(x-1)^2}[/tex]
Now substitute x = 6
[tex]\frac{dy}{dx}|_{x=6}= \frac{-5000}{(6-1)^2}[/tex]
[tex]\frac{dy}{dx}|_{x=6}= \frac{-5000}{5^2}[/tex]
[tex]\frac{dy}{dx}|_{x=6}= \frac{-5000}{25}[/tex]
[tex]\frac{dy}{dx}|_{x=6}= -200[/tex]
Hence [tex]\frac{dy}{dx}|_{x=6}= -200[/tex] =T-shirts per dollar
