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

First derivative: dy/dx = $\frac{dy}{dt}$ / $\frac{dx}{dt}$ = g'(t) / f'(t)

Second derivative: $d^{2y}$/$dx^2$ = \frac{d}{dx}$$\frac{dy}{dx} = [d/dt$\frac{dy}{dx}$] / $\frac{dx}{dt}$

WRONG formula: $d^{2y}$/$dx^2$ ≠ $\frac{d^2y/dt^2}{(d^2x/dt^2)}$. 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^$\frac{2}{2t}$ = 3t/2
• d/dt$\frac{3t}{2}$ = 3/2
• $d^{2y}$/$dx^2$ = $\frac{3/2}{(2t)}$ = $\frac{3}{4t}$

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 = -$\frac{b}{a}$*cot t
• Parabola: x = $at^2$, y = 2at -> dy/dx = 1/t
• Astroid: x = a*$cos^3$ t, y = a*$sin^3$ t -> dy/dx = -tan t