Han Song has deleted his previous answer.

The correct working is :

y = (11x - 1)/(9 - x)

dy/dx = [(9 - x)(11) - (11x - 1)(-1)] / (9 - x)²

(Quotient rule)

= 11/(9 - x) + (11x - 1)/(9 - x)²



Since y increases at a constant rate of 0.005 units/s from an initial of 2.9 (where t = 0),

dy/dt = 0.005 (a constant)

After 20 seconds,

y = 2.9 + 0.005 x 20
= 2.9 + 0.1
= 3

(you can see this as though it's a linear graph, where y = dy/dt (t) + c, where c = 2.9 and dy/dt is constant 0.005)

When y = 3,

3 = (11x - 1)/(9 - x)

3(9 - x) = 11x - 1

27 - 3x = 11x - 1

14x = 28

x = 2


So at x = 2, dy/dx

= 11/(9 - 2) + (11(2) - 1)/(9 - 2)²

= 11/7 + 21/49

= 77/49 + 21/49

= 98/49

= 2



So after 20 seconds, (where y = 3 and x = 2)

dx/dt = dy/dt x dx/dy

= dy/dt ÷ dy/dx

= 0.005 ÷ 2

= 0.0025

So rate of change of x after 20 seconds
= 0.0025 units/s

(Exact)

Edited

Hi J, thank you!:)

Welcome

Another way to differentiate:

y = (11x - 1)/(9 - x)

= (11x - 99 + 98)/(9 - x)

= 98/(9 - x) - 11(9 - x)/(9 - x)

= 98/(9 - x) - 11


dy/dx = 98(-1)(-1)/(9 - x)²
= 98/(9 - x)²

This is actually more simplified.


Then when x = 2,

dy/dx = 98/(9 - 2)²

= 98/49

= 2

Weird, I have a different answer...

Edit: I went to multiply

- (11x - 1) (-1)

and stupidly wrote + 11x - 11.