Question

Find `dy/dx if x + sqrt(xy) + y = 1`

Answer 1

`x + sqrt(xy) + y = 1`

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

`1 + 1/(2sqrt(xy)). d/dx (xy) + dy/dx = 0`

∴ `1 + 1/(2sqrt(xy)).[x dy/dx + y × 1] + dy/dx = 0`

∴ `1 + 1/2 sqrt(x/y) dy/dx + (1)/(2)sqrt(y/x) + dy/dx = 0`

∴ `(1/2 sqrt(x/y) + 1) dy/dx = −(1)/(2)sqrt(y/x) - 1`

∴ `((sqrt(x) + 2sqrt(y))/(2sqrt(y))) dy/dx = (-sqrt(y) -2sqrt(x))/(2sqrt(x)`

∴ `dy/dx = (-sqrt(y) -2sqrt(x))/(cancel2sqrt(x)) × (cancel2sqrt(y))/((sqrt(x) + 2sqrt(y))`

∴ `dy/dx = (-sqrt(y)(2sqrt(x) + sqrt(y)))/(sqrt(x)(sqrt(x) + 2sqrt(y))`.