Q.If $x \sqrt{1+y} + y \sqrt{1+x} = 0$, for $-1 < x < 1$, prove that $\frac{dy}{dx} = -\frac{1}{(1+x)^2}$

Idea: the given relation hides a simple $y=y(x)$; find it, then differentiate.

From $x\sqrt{1+y} + y\sqrt{1+x} = 0$ we have $x\sqrt{1+y} = -y\sqrt{1+x}$. Squaring:

The branch $x=y$ satisfies the original equation only at $x=0$, so the valid relation is …

Rather than differentiate the messy square-root equation directly, we first uncover the simple relationship it hides.

Step 1 — rewrite and square

Square both sides (we will check for extraneous roots afterward):

Step 2 — factor

So either $x-y=0$ or $x+y+xy=0$.

Step 3 — pick the correct branch

If $y=x$, the original equation becomes $2x\sqrt{1+x}=0$, true only at $x=0$ — not a valid relation on the whole interval, so it is extraneous from squaring. Therefore we take

$x+y+xy = 0 \;\Rightarrow\; y = -\frac{x}{1+x}.$ …

Method: Simplify the Implicit Relation Before Differentiating

Sometimes an implicit relation looks intimidating (square roots, products) but actually hides a much simpler relationship between $x$ and $y$. Uncovering that simpler form first — by algebraic manipulation, not calculus — makes the differentiation step trivial.

Steps

Step 1: Manipulate the given relation algebraically (isolate a square root, then square) to find a simpler equivalent relation

Step 2: Factor the resulting polynomial relation to find candidate simplified forms

Step 3: Test each candidate against the ORIGINAL (unsquared) equation to discard any extraneous branch introduced by squaring

Step 4: Differentiate the genuine simplified relation using ordinary rules (quotient rule, etc.) …

Common Mistakes

Mistake 1: Differentiating the original square-root equation directly via implicit differentiation, without simplifying first.

Why it's wrong: this is technically possible but produces a much messier expression full of nested square roots — the algebraic simplification route is dramatically cleaner and less error-prone for this particular relation. Correct approach: always check whether an implicit relation simplifies algebraically before committing to direct implicit differentiation.

Mistake 2: Accepting both branches from the squared equation ($x=y$ and $x+y+xy=0$) without testing them against the original equation. …