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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)`
Solution
`-(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`.
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