AP Calc AB: If 2+2sinx=3tany and dx/dt=2, find dy/dt when x=pi/6 and y=pi/4.
(Answer is 1/root3 but I need the work of how to reach the answer).

Differentiate both sides of

2 + 2 sin(x) = 3 tan(y)

with respect to a new variable t, assuming both x and t depend on t, which will involve the chain rule:

d(2 + 2 sin(x))/dt = d(3 tan(y))/dt

d(2)/dt + d(2 sin(x))/dt = 3 d(tan(y))/dt

0 + 2 d(sin(x))/dt = 3 sec²(y) dy/dt

2 cos(x) dx/dt = 3 sec²(y) dy/dt

Now, when x = π/6, y = π/4, and dx/dt = 2, solve for dy/dt :

2 cos(π/6) × 2 = 3 sec²(π/4) dy/dt

dy/dt = 4 cos(π/6) / (3 sec²(π/4))

dy/dt = 4 (√3/2) / (3 (√2)²)

dy/dt = √3/3 = 1/√3

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