Question

Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt.

xy = 2

a. Find dy/dt, when x = 4, given that dx/dt = 13.
b. Find dx/dt, when x = 1, given that dy/dt = -9.

Answer 1

a.
`(dy)/(dt) = -(13)/(8)`

b.
`(dx)/(dt) = (9)/(2)`

Explanation:

To solve this question, we apply implicit differentiation.

xy = 2

Applying the implicit differentiation:

`y(dx)/(dt) + x(dy)/(dt) = (d)/(dt)(2)`

`y(dx)/(dt) + x(dy)/(dt) = 0`

a. Find dy/dt, when x = 4, given that dx/dt = 13.

x = 4

So

`xy = 2`

`4y = 2`

`y = (2)/(4) = (1)/(2)`

Then

`y(dx)/(dt) + x(dy)/(dt) = 0`

`(1)/(2)(13) + 4(dy)/(dt) = 0`

`4(dy)/(dt) = -(13)/(2)`

`(dy)/(dt) = -(13)/(8)`

b. Find dx/dt, when x = 1, given that dy/dt = -9.

x = 1

So

`xy = 2`

`y = 2`

Then

`y(dx)/(dt) + x(dy)/(dt) = 0`

`2(dx)/(dt) - 9 = 0`

`2(dx)/(dt) = 9`

`(dx)/(dt) = (9)/(2)`