Graphing a non function derivative

Question

I'm in the midst of a graphing project, and I wanted to make a loop with a non-function because those can make some pretty cool shapes (EC). I made a loop, found the derivative, but Desmos is giving me hard time graphing the derivative. It seems impossible to isolate either x or y, so I'm not quite sure how to graph it, may not be possible in a program such as desmos. Here are my equations and picture of the original equation.

$$y=(x^4)/2+y^2/x$$ $$dy/dx=(2x^3-y^2x^{-2})/(1-2yx^{-1})$$

Original Equation

Answer 1

It's not impossible to isolate $y$. Notice the quadratic in $y$. Now apply the quadratic formula.

Answer 2

Isolate y with the quadratic formula: $$y={x\pm x \sqrt{1-2x^3} \over 2}$$ Substitute into the derivative and simplify: $$y=\frac 1 2 \pm {5x^3-1 \over 2 \sqrt{1-2x^3}}$$ Pretty fun problem!