I know that $|x|$ is not differentiable at $x=0$ because there is potentially an infinite number of tangent lines going through that point.
But let's say we were interested in the motion of an object through time starting at the point $0$ and going forward, wouldn't we then be able to talk about the instantaneous rate of change of $y$ with regard to $x$ at $x=0$? And wouldn't it be 1?
If not, how would we be able to talk about the rate of change of position to time at the point $0$?
I'm having trouble reconciling those two ideas; namely that there seems to be a rate of change at $0$ if we specify a direction on the $x$ axis, while at the same time the derivative technically doesn't exist at that point.