Final answer:
Differentiating the equation xy = -1 with respect to time and substituting x = -3 and dy/dt = 1, we find that dx/dt = 9. This is the rate of change of x with respect to time when x is -3.
Step-by-step explanation:
The student has provided the equation xy = -1 and the rate of change dy/dt = 1. To find dx/dt when x = -3, we need to differentiate the given equation with respect to time t.
Using the product rule, we get:
d/dt(xy) = d/dt(-1)
x(dy/dt) + y(dx/dt) = 0
-3(1) + y(dx/dt) = 0, since dy/dt = 1 and x = -3.
Now we have to find the value of y when x = -3. From the equation xy = -1, we can deduce that:
y = -1/x
y = -1/(-3)
y = 1/3
Substituting y = 1/3 back into the equation:
-3(1) + (1/3)(dx/dt) = 0
(1/3)(dx/dt) = 3
dx/dt = 9.
Therefore, the rate of change of x with respect to time t when x is -3 is 9 units per time unit.