I'm in the middle of my first calculus class, a week ago we covered how to find the derivative of implicit functions, and I'm still thinking about it. I completely understand what I am supposed to understand about the topic—it's sort of a small part of calculus 1 that I guess isn't relevant at this level—but functions that ignore the vertical line test are incredibly interesting to me, and I want to know what the derivative looks like, not just how to plug an x and y value and get the derivative of just one point on the graph.
Let's take $y^3+x^3=3xy$
The derivative is $\frac{y-x^{2}}{y^{2}-x}$ , you plug in the x and y values from the original function to find the tangent slope of that point, easy enough, but what does it look like? You can't graph just $\frac{y-x^{2}}{y^{2}-x}$ on a 2D grid, so I tried setting it to a number. I chose easy to work with numbers like -1, 0, 1, that would hopefully illuminate useful and easy to understand slopes.
A graph of $y^3+x^3=3xy$, $\frac{y-x^{2}}{y^{2}-x}=-1$, and y=-x-3
A graph of $y^3+x^3=3xy$, $\frac{y-x^{2}}{y^{2}-x}=0$, and $\frac{y-x^{2}}{y^{2}-x}=1$
(I forgot to simplify in desmos but it's the same, sorry)
The properties of these functions were immediately visually obvious to me, the derivative function, when set to an arbitrary number, will intersect the original function at every point where the slope is equal to that arbitrary number. The slope is -1 at the point (1.5, 1.5), exactly where $\frac{y-x^{2}}{y^{2}-x}=-1$ intersects $y^3+x^3=3xy$. And curiously, it seems to perfectly match the slant asymptote of y=-x-1, where I would presume the derivative limited to infinity would be -1 (I don't know how to say that in jargon or with notation, sorry). It also seems to fit that all of these functions have a hole at (0,0), which makes sense, because the slope at (0,0) is clearly not -1. or 1. But, also curiously $\frac{y-x^{2}}{y^{2}-x}=0$ also has a hole at the origin, even though it looks like one of two slopes at the origin is clearly 0, and in fact limits to 0. Very interesting.
I asked my professor about this and he explained that you can't really do this. And that I'll learn how to do something like it, a slope field, next quarter in calculus 2.
But I'm still not exactly satisfied—yes looking at the slope field graph it makes sense to me, that's a pretty good approximation of the slopes of all x and y values for the function—but surely there has to be a way to simply graph one or several functions of x that tell you one or several possible slopes for each value of x on the graph. I can intuitively look at the graph and see that it's shape is continuous, and I can even just plot a bunch of different values for the derivative (y) in terms of x, and then crudely connect them together exactly like this.
Theorized 2D derivative of $y^3+x^3=3xy$
With vertical asymptotes that break the continuity of the derivative wherever the slope is just a flat x=n.
Or like this for $2\left(x^{2}+y^{2}\right)^{2}=25\left(x^{2}-y^{2}\right)$ (ignore the ellipse in red, I was just using that to help orient my drawing)
Theorized 2D derivative of $2\left(x^{2}+y^{2}\right)^{2}=25\left(x^{2}-y^{2}\right)$
But how do I actually find the definition of these functions? Surely it's possible with arithmetic right? How would you even define a... triple loop-de-loop infinity sign thing? What's that shape even called? Or would it just be two separate circles and an ellipse?
Am I onto something? Is this something that's already been made rigorous? If so, what is it called? or am I inventing my own thing here? I can't find anything online about it and my professor doesn't seem to know either. I made an account here just to talk about this. Is this a question that doesn't get answered because it's something you can only really ask without an understanding of complex planes or integrals or whatever else basic topics I just simply have yet to learn about in mathematics?