If you look at a function "infinitely close", the difference between two points is a line:
__
__/
__/
Where each "__" is a point, and "/" is the value of a derivative (assume the two "/"s have different slopes). I have this intuition because a function can be approximated by a line at an infinitely close distance (i.e. "Linear Approximations")
If you use the above graph of the function to graph the derivative, the derivative looks like this:
_
_|
So at an infinitely close distance the derivative looks like the second graph above.
But now if you look at the derivative at an infinitely close distance, it looks the the first graph. So how can the derivative look two different ways at an infinitely close distance? It looks like the first graph when you look at it directly at an infinitely close distance, but the second graph if you look at its antiderivative at an infinitely close distance and use that to plot it.
I know this obviously isn't rigourous but what part of my intuition is wrong?