Would $f(x,y) = \frac{y}{x}$ give an actual graph of all the possible slopes of a function of 1 variable?

Question

If you let $y$ stand for $\Delta y$ and $x$ stand for $\Delta x$.

I wanted to plot this because I thought it would be interesting among other things to see the behavior of the function around $x=0$.

I'm also assuming that this function would show you the graph of all the possible derivatives of functions of 1 variable.

Sorry if this is too low-level or completely obvious. It seems kind of obvious even to me, but I've often been wrong with math when things seemed completely obvious in the past...

Answer 1

Yes. Look up "slope field".

For example, here:

https://en.wikipedia.org/wiki/Slope_field