<s>[INST] Points on $(x^2 + y^2)^2 = 2x^2 - 2y^2$ with slope of $1$ Let the curve in the plane defined by the equation: $(x^2 + y^2)^2 = 2x^2 - 2y^2$
How can i graph the curve in the plane and determine the points of the curve where $\frac{dy}{dx} = 1$.
My work:
First i found the roots of this equation with a change of variable $z = y^2$ and get: 

and then i tried to graph the point $ x - y$ and $x + y $ but i stuck i can't graph this and find the point  where the derivative is 1. Some help please.

 [/INST] You've already got some good suggestions on the graphing part using polar coordinates (if you are familiar with that). For the other, implicitly differentiate the given relation to obtain ${dy\over dx}$:
\begin{align*}
(x^2 + y^2)^2 &= 2x^2 - 2y^2\\
2(x^2+y^2)(2x+2y{dy\over dx})&=4x-4y{dy\over dx}\\
2x^3+2x^2y{dy\over dx}+2xy^2+2y^3{dy\over dx}&=2x-2y{dy\over dx}\\
(2x^2y+2y^3+2y){dy\over dx}&=2x-2x^3-2xy^2\\
{dy\over dx}&={2x-2x^3-2xy^2\over 2x^2y+2y^3+2y}\\
{dy\over dx}&={x(1-x^2-y^2)\over y(x^2+y^2+1)}\\
\end{align*}
Now solve ${dy\over dx}=1$ in terms of $x$ and $y$, keeping in mind you can use the original expression to simplify things along the way. Edit: This was much more involved than I anticipated; see below.
Here's the situation (note ${dy\over dx}$ is not defined at the origin):

So how to find these coordinates? As noted in another answer, the original relation can be expressed in polar coordinates as $r^2=2\cos(2t)$. Recall that the slope of the tangent line in polar coordinates is given by 
$$
{dy\over dx}={{dr\over dt}\sin t+r\cos t\over {dr\over dt}\cos t-r\sin t},
$$
so setting ${dy\over dx}=1$ and using $r^2=2\cos(2t)\implies {dr\over dt}={-2\sin(2t)\over r}$, we get
\begin{align*}
{dy\over dx}&=1\\
{dr\over dt}\sin t+r\cos t&={dr\over dt}\cos t-r\sin t\\
{-2\sin(2t)\over r}\sin t+r\cos t&={-2\sin(2t)\over r}\cos t-r\sin t\\
-2\sin(2t)\sin t+r^2\cos t&=-2\sin(2t)\cos t-r^2\sin t\\
r^2&={2\sin(2t)[\sin t-\cos t]\over \cos t+\sin t}\\
2\cos(2t)&={2\sin(2t)[\sin t-\cos t]\over \cos t+\sin t}\\
\end{align*}
Multiply on the right by ${\cos t-\sin t\over \cos t-\sin t}$, use $\cos^2t-\sin^2t=\cos(2t)$ and $\sin(2t)=2\sin t\cos t$ and rearrange to get 
\begin{align*}
\sin(2t)&=-(\sin t+\cos t)^2\\
2\sin t\cos t&=-1-2\sin t\cos t\\
4\sin t\cos t&=-1\\
\sin(2t)&=-{1\over 2}\\
t&={7\pi\over 12},\ {11\pi\over 12}.
\end{align*}
But $r^2=2\cos(2t)$ so, $t={11\pi/12}\implies r^2=2\cos(11\pi/6)=\sqrt{3}$. Thus, the polar coordinates of this point are $(3^{1/4},11\pi/12)$ which in rectangular coordinates is $$(3^{1/4}\cos(11\pi/12),3^{1/4}\sin(11\pi/12))\approx (-1.27,0.34)$$ as shown in the figure above.
(The fourth quadrant point of tangency can be found similarly.)
</s>