xy=-2 and dy/dt=4. Find dx/dt when x=1.
xy = -2
x dy/dt + y dx/dt = 0
using the given:
when x = 1, y = -2
1(4) + (-2)(dx/dt) = 0
dx/dt = -4/-2 = 2
To find dx/dt when x = 1, we need to differentiate both sides of the given equation xy = -2 with respect to t (the independent variable, typically representing time).
Applying the product rule of differentiation, we have:
d/dt(xy) = d/dt(-2)
To differentiate xy with respect to t, we can use the chain rule:
d(xy)/dt = x(dy/dt) + y(dx/dt)
Now, let's substitute the given values: dy/dt = 4 and x = 1.
Therefore, the equation becomes:
1(dy/dt) + y(dx/dt) = 0
Substituting dy/dt = 4, we have:
1(4) + y(dx/dt) = 0
Simplifying, we get:
4 + y(dx/dt) = 0
Now, we need to find the value of y when x = 1. From the given equation xy = -2, we can solve for y:
1 * y = -2
y = -2
Substituting y = -2, we have:
4 + (-2)(dx/dt) = 0
Simplifying further:
4 - 2(dx/dt) = 0
To find dx/dt, isolate the term containing it:
-2(dx/dt) = -4
Finally, solve for dx/dt:
dx/dt = (-4) / (-2)
dx/dt = 2
Therefore, when x = 1, dx/dt = 2.