dy / dx

[Answering the question is dy/dx a ratio?] Leibniz's notation is very suggestive and very useful; even though derivatives are not really quotients, in many ways they behave as if they were quotients [e.g the Chain Rule). So, even though we write dy/dx as if it were a fraction, and many computations look like we are working with it like a fraction, it isn't really a fraction (it just plays one on television). 

[But] the notation is so nice and so suggestive, we keep the notation even though the notation no longer represents an actual quotient, it now represents a single limit. In fact, Leibniz's notation is [so useful] and better than Newton's notation that England fell behind all of Europe for centuries in mathematics and science because, due to the fight between Newton's and Leibniz's camp over who had invented Calculus and who stole it from whom (consensus is that they each discovered it independently), England's scientific establishment decided to ignore what was being done in Europe with Leibniz notation and stuck to Newton's... and got stuck in the mud in large part because of it [..] It wasn't until [the mathematician] Hamilton that they started coming back, and when they reformed math teaching in Oxford and Cambridge, they adopted the continental ideas and notation. Hah! This is so incredibly hard to believe.. England fell behind because of some stupid notation? And am I trying to make an underhanded comment on Brexit here? Maybe I am ! :)

Note: the excerpts above in no way subtracts from Newton's research, his insights into mathematics. The man was simply a giant, the originator of many, many groundbreaking new ideas.