Question

Suppose xy=−4 and dy/dt=−3. Find dx/dt when x=−1.

Answer 1

By using implicit differentiation and substituting the given values, we find that dx/dt equals -0.75 when x is -1.

Step-by-step explanation:

The student has provided the information that xy = -4 and dy/dt = -3, and is asking to find dx/dt when x = -1. To solve this, we use implicit differentiation on the first equation. Differentiating both sides of xy with respect to t gives:

d(xy)/dt = d(-4)/dt → x(dy/dt) + y(dx/dt) = 0

Substituting the known values, we have:

-1(-3) + y(dx/dt) = 0 → 3 + y(dx/dt) = 0

Since xy = -4, when x = -1, y = 4. Therefore, we substitute y into the equation:

3 + 4(dx/dt) = 0

Solving for dx/dt, we get:

dx/dt = -3/4 → dx/dt = -0.75

Answer 2

When x=-1:
`\quad (-1)y=-4\qquad\to\qquad y=4\)`

Ok that gives us a little more information.
If we implicitly differentiate with respect to t, from the very start, then we can apply our product rule, ya?


`x'y+xy'=0`

The right side is zero, derivative of a constant is zero.
Where x' is dx/dt and y' is dy/dt.

From here, plug in all the stuff you know:
y' = -3
x = -1
y = 4

and solve for x'.

Hope that helps!