When x = f(t) and y = g(t):

First derivative: dy/dx = (dy/dt) / (dx/dt) = g'(t) / f'(t)

Second derivative: d^2y/dx^2 = d/dx(dy/dx) = [d/dt(dy/dx)] / (dx/dt)

WRONG formula: d^2y/dx^2 ≠ (d^2y/dt^2)/(d^2x/dt^2). This is a common JEE trap.

Correct procedure for d^2y/dx^2:

1. Find dy/dx as a function of t
2. Differentiate dy/dx with respect to t
3. Divide by dx/dt

Example: x = t^2, y = t^3

• dy/dx = 3t^2/(2t) = 3t/2
• d/dt(3t/2) = 3/2
• d^2y/dx^2 = (3/2)/(2t) = 3/(4t)

Common parametric curves in JEE:

• Circle: x = acos t, y = asin t -> dy/dx = -cot t
• Ellipse: x = acos t, y = bsin t -> dy/dx = -(b/a)*cot t
• Parabola: x = at^2, y = 2at -> dy/dx = 1/t
• Astroid: x = acos^3 t, y = asin^3 t -> dy/dx = -tan t