I have a test tomorrow and I am doing homework to review and study. The problem is to differentiate $$ x^2 (x-y)^2 = x^2 - y^2. $$ I tried multiple times; however, every time I try I get the incorrect answer. It would be greatly appreciated if somebody could show me the steps on how to get the answer. (No full answer please; I still want to learn).
The answer according to the book is $$ \frac{-2x^3 + 3x^2y - xy^2 + x}{x^y - x^3 + y}. $$
$\frac{d}{dx}[x^2(x-y)^2 = x^2 - y^2]$
$\Rightarrow 2x(x-y)^2 + x^2\cdot 2(x-y) \cdot (1-\frac{dy}{dx}) = 2x - 2y\cdot\frac{dy}{dx}$
$\Rightarrow 2x^3-4x^2y + 2xy^2 +2x -2x\frac{dy}{dx}-2y + 2y\frac{dy}{dx}= 2x -2y\frac{dy}{dx}$
$\Rightarrow 2x^3 - 4x^2y + 2xy^2 - 2y = (2x-4y)\frac{dy}{dx}$
$\frac{dy}{dx}=\frac{x^3 - 2x^2y+xy^2 -y}{x-2y}$