find dy/dx if x+y/x-y=x^2+y^2
(x + y)(x – y)^(-1) = x^2 + y^2
Differentiate the lhs, wrt x, using the product rule,
(1 + y’)(x - y)^(-1) + (x + y)(-1)(x – y)^(-2)*(1 – y’) = 2x + 2y.y’
(x - y)^(-1) + y’. (x - y)^(-1) – (x + y)(x – y)^(-2) + y’.(x + y)(x – y)^(-2) = 2x + 2y.y’
(x - y)^(-1) – (x + y)(x – y)^(-2) + y’.(x - y)^(-1) + y’. (x + y)(x – y)^(-2) = 2x + 2y.y’
{(x - y) – (x + y)} / (x – y)^(-2) + y’{(x - y)^(-1) + (x + y)(x – y)^(-2)} = 2x + 2y.y’
{-2y)} / (x – y)^(-2) + y’{[(x - y) + (x + y)] / (x – y)^(-2)} = 2x + 2y.y’
{-2y)} / (x – y)^(-2) + y’{[2x] / (x – y)^(-2)} = 2x + 2y.y’
-y / (x – y)^(-2) + y’.x / (x – y)^(-2) = x + y.y’
-y / (x – y)^(-2) – x = -y’.x / (x – y)^(-2) + y.y’
-y – x(x – y)^2 = -y’.x + y.y’(x – y)^2
-y – x^3 + 2x^2.y – xy^2 = y’(-x + x^2.y – 2xy^2 + y^3)
dy/dx = y’ = (-y – x^3 + 2x^2.y – xy^2) / (-x + x^2.y – 2xy^2 + y^3)