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Question
If `xsqrt(1+y) + y sqrt(1+x) = 0`, for, −1 < x < 1, prove that `dy/dx = -1/(1+ x)^2`.
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Solution
`x sqrt(1 + y) + y sqrt(1 + x) = 0`
∴ `xsqrt(1 + y) = - y sqrt(1 + x) = 0`
On squaring both sides,
x2 (1 + y) = y2 (1 + x)
⇒ x2 + x2y = y2 + y2x
⇒ x2 – y2 – y2x + x2y = 0
⇒ (x – y)(x + y) + xy(x – y) = 0
⇒ (x – y)[x + y + xy] = 0
x – y = 0 ⇒ x ≠ y
x + y (1 + x) = 0
∴ y = `-x/(1 - x)`
∴ `dy/dx = ((1 + x)(1) - x * 1)/(1 + x)^2`
= `-(1 + x - x)/(1 + x)^2`
= `-1/(1 + x)^2`
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