Question

Suppose xy=4 and dy/dt=-2
Find dx/dt when x=-3

Answer 1

First, find the derivative of both sides of the equation xy=4 with respect to t. Any idea how to do that?

Answer 2

For just the derivative of y?

Answer 3

What we need to do is something called "implicit differentiation". Is that a topic that rings a bell? Are you all doing that now?

Answer 4

Yeah, it's what I'm learning but I'm confused

Answer 5

You're used to taking the derivative of y with respect to x (dy/dx). Now we're going to introduce another variable, t. and take the derivative of both sides with respect to t. Don't let that confusing babble confuse you, though. It's not that bad

Answer 6

To find the derivative of xy (the left side of the equation) with respect to t, we just use the product rule.
x times (derivative of y with respect to t) + (derivative of x with respect to t) times y
i'm sure the product rule rings a bell.

Answer 7

so the left side will look like:
x(dy/dt) + y(dx/dt)

Answer 8

on the right side, it's easy.
the derivative of 4 with respect to t is just 0.

Answer 9

So...after taking the derivative of both sides with respect to t, we get
x(dy/dt) + y(dx/dt) = 0

Answer 10

Okay, I get that

Answer 11

for your problem, you want to find dx/dt. You know dy/dt and x, because they're given in your problem. You can get y by plugging x = -3 into xy = 4 and solving for y. You then have all the pieces you need to plug in the formula we came up with and solve for dx/dt.

Answer 12

I'm not sure if it's right but after I plug it in I get 9/2

Answer 13

This is my way of doing this:

dx/dt = dx/dy * dy/dt (dy cancelled out)

given info: dy/dt= -2
xy = 4 => y = 4/x => dy/dx = -4/x^2.
Since x = -3 => dy/dx = -4/9

=> dx/dt = dx/dy * dy/dt
=> dx/dt = -4/9 * -2
=> dx/dt = 8/9