Assume x and y are functions of t. Evaluate dy/dt.
xy - 5x + 2x^3 = -70; dx/dt=-5, x=2, y=-3
xdy/dt + ydx/dt - 5dx/dt + 6x^2dx/dt = 0
2dy/dt - 2(-5) - 5(-5) + 6(25)(-5) = 0
solve for dy/dt
To evaluate dy/dt, we need to differentiate the equation with respect to t and isolate dy/dt.
Given equation: xy - 5x + 2x^3 = -70
First, let's differentiate both sides of the equation with respect to t using the product rule and chain rule:
d/dt (xy) - d/dt (5x) + d/dt (2x^3) = d/dt (-70)
Next, we need to apply the chain rule to the terms that involve x and y. Since both x and y are functions of t, we need to multiply by their respective derivatives:
[x(dy/dt) + y(dx/dt)] - [5(dx/dt)] + [6x^2(dx/dt)] = 0
Substituting the given values for dx/dt, x, and y:
[2(dy/dt) + (-3)(-5)] - [5(-5)] + [6(2)^2(-5)] = 0
Simplifying the equation:
(2(dy/dt) + 15) + 25 + (-60) = 0
2(dy/dt) - 20 = 0
Now, isolate dy/dt:
2(dy/dt) = 20
dy/dt = 20/2
dy/dt = 10
Therefore, the value of dy/dt is 10.