If xy^2=12 and dy/dt=6, find dx/dt when y=2

[dcgirl16]

If xy^2=12 and dy/dt=6, find dx/dt when y=2

The way i thought to do this would be
(1)(y^2)+x(y)(dy/dt)=0 but i don't know x so this isn't working, what am i doing wrong

 

[Terilien]

You just multiply the derivative of x with respect to y by 6(chain rule). You can find x as a function of y by solving for it so x=12/y^2.

 

[tacman]

?

To solve this problem, we can use implicit differentiation. We have the equation xy^2 = 12, so we can differentiate both sides with respect to t:

d/dt (xy^2) = d/dt (12)

Using the product rule on the left side, we get:

y^2 (dx/dt) + x(2y)(dy/dt) = 0

Substituting in the given value for dy/dt = 6 and the given value for y = 2, we have:

(2)^2 (dx/dt) + x(2)(6) = 0

Simplifying, we get:

4(dx/dt) + 12x = 0

Now, we can solve for dx/dt by isolating it on one side:

4(dx/dt) = -12x

dx/dt = (-12x)/4

dx/dt = -3x

Since we are given the value of y = 2, we can substitute it into the original equation xy^2 = 12 to solve for x:

x(2)^2 = 12

4x = 12

x = 3

Now, we can finally solve for dx/dt when y = 2 and x = 3:

dx/dt = -3(3) = -9

Therefore, when y = 2, dx/dt = -9.