Question

If `xsqrt(1+y) + y  sqrt(1+x) = 0`, for, −1 < x < 1, prove that `dy/dx = -1/(1+ x)^2`.

If `xsqrt(1 + y) + ysqrt(1 + x) = 0, -1 < x < 1, x ≠ y,` then prove that `(dy)/(dx) = (-1)/(1 + x)^2`.

Answer 1

`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

As x ≠ y

x + y (1 + x) = 0

y (1 + x) = –x

∴ y = `-x/(1 - x)`

Differentiating w.r.t., x

∴ `dy/dx = ((1 + x)(1) - x (1))/(1 + x)^2`

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

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