Question

Choose the correct option from the given alternatives :

If `xsqrt(y + 1) + ysqrt(x + 1) = 0 and x ≠ y, "then" "dy"/"dx"` = ........

Options

• `(1)/(1 + x)^2`

• `-(1)/(1 + x)^2`

• (1 + x)² 

• `-x/(x + 1)`

Answer 1

`-(1)/(1 + x)^2`

Explanation:

`xsqrt(y + 1) = -ysqrt(x + 1)`

Squaring both the sides,

∴ x²(y + 1) = y²(x + 1)

∴ x²y + x² = xy² + y²

∴ x² – y² = xy² – x²y

∴ (x – y)(x + y) = – xy(x – y)

∴ x + y = – xy                      ...[∵ x ≠ y] 

∴ x = – xy – y

∴ x = – y (x + 1)

∴ y = `- x/(x + 1)`

Differentiating both sides w.r.t.x, we get

`dy/dx = - [(1 + x) d/dx(x) - (x) d/dx (x + 1)]/(1 + x)^2`

`dy/dx = - [(1 + x). 1 - x(1 + 0)]/(1 + x)^2`

`dy/dx = - [1 + cancelx - cancelx]/(1 + x)^2`

∴ `"dy"/"dx" = -(1)/(1 + x)^2`.