Calculus Question

Question

Assume x and y are functions of t. Evaluate dy/dt for 2xy-4x+4y^3=-28, with the conditions dx/dt=-12, x=4, y=-1.

dy/dt=?

Answer 1

2xy - 4x + 4y³ = -28

Confirm that x = 4, y = -1 gives a true statement:

8(-1) - 4(4) + 4(-1) = -8 - 16 - 4 = -28.

Using implicit differentiation with respect to t, apply product rule to first term:

2x (dy/dt) + y(2) (dx/dt)

2nd term:

-4 (dx/dt)

3rd term:

12y²(dy/dt)

That gives:

2x(dy/dt) + 2y(dx/dt) -4(dx/dt) + 12y²(dy/dt) = 0 (derivative of -28 is 0)

At this point we could solve for dy/dt in terms of x, y and dx/dt, but easier to substitute the given info, then solve:

2(4)(dy/dt) + 2(-1)(-12) -4(-12) + 12(-1²)(dy/dt) = 0

20(dy/dt) + 24 +48 = 0

20(dy/dt) = -72

dy/dt = -72/20 = -18/5