Assume x and y are functions of t. Evaluate dy/dt for 2xy-2x+3y^3=-15, with the conditions dx/dt=-12, x=3, y=-1.
To find dy/dt, we first differentiate both sides of the equation 2xy - 2x + 3y^3 = -15 with respect to t using the product rule and chain rule.
d(2xy)/dt - d(2x)/dt + d(3y^3)/dt = d(-15)/dt
2*(dx/dt)*y + 2x*(dy/dt) - 2*(dx/dt) + 9y^2*(dy/dt) = 0
Substitute the given value of dx/dt = -12 into the equation:
2*(-12)*(-1) + 2*3*(dy/dt) -2*(-12) + 9(-1)^2*(dy/dt) = 0
24 + 6*(dy/dt) + 24 - 9*(dy/dt) = 0
-3*(dy/dt) = -48
dy/dt = -48 / -3
dy/dt = 16
Therefore, dy/dt = 16.