# How do you differentiate y=(x-y)^2/(x+y)?

Sep 25, 2017

Given: y=(x-y)^2/(x+y); x!=-y

Multiply both sides by $x + y$:

$x y + {y}^{2} = {\left(x - y\right)}^{2}$

Expand the square:

$x y + {y}^{2} = {x}^{2} - 2 x y + {y}^{2}$

Combine like terms:

$3 x y = {x}^{2}$

Divide both sides by $3 x$:

$y = \frac{1}{3} x$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{3}$

If you do not believe the result, here is a graph of $y = {\left(x - y\right)}^{2} / \left(x + y\right)$ to prove that is merely a line with a slope of $\frac{1}{3}$: