Method of Differentiation
Implicit Differentiation
IIT-JEE 1983
Grade 12
Question:
If $x\sqrt{1 + y} + y\sqrt{1 + x} = 0$ for $-1 < x, y < 1$, $x \neq y$, then $\frac{dy}{dx}$ equals:
(1) $\frac{1}{(1 + x)^2}$
(2) $-\frac{1}{(1 + x)^2}$
(3) $\frac{1 + y}{1 + x}$
(4) $-\frac{1 + y}{1 + x}$
Step-by-Step Solution
Key Concept: Rearrange and square both sides to show the relation simplifies to $x + y + 2xy = 0$ (after cancelling the factor $x - y$), which is far easier to differentiate implicitly.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)