Decoding Leibniz Notation (2024)

Summary

Decoding Leibniz notation I wrote this for myself to understand the Leibniz notation. Prerequisites for this post are the definition of the derivative and the Lagrange notation. If you don’t understand these yet, please study them first. So… You may have already seen something like dydxdxdy​. This is called the Leibniz notation. The Leibniz notation has many of what Spivak calls “vagaries”. It has multiple interpretations– formal and informal. The informal interpretation doesn’t map to modern mathematics, but can sometimes be useful (while at other times misleading). The full, unambiguous Leibniz notation is verbose, so in practice people end up taking liberties with it. As a consequence, its meaning must often be discerned from the context. This flexibility makes the notation very useful in science and engineering, but also makes it difficult to learn. I explore it here to make learning easier. Historical motivation We start with the historical interpretation, where the notation began. Leibniz didn’t know about limits. He thought the derivative is the value of the quotient f(x+h)−f(x)hhf(x+h)−f(x)​ when hhh is “infinitesimally small”. He denoted this infinitesimally small quantity of hhh by dxdxdx, and the corresponding difference f(x+dx)−f(x)f(x+dx)-f(x)f(x+dx)−f(x) by df(x)df(x)df(x). Thus for a given function fff the Leibniz notation for its derivative f′f’f′ is: df(x)dx=f′dxdf(x)​=f′ Intuitively, we can think of ddd in a historical context as “delta” or “change”. Then we can interpret this notation as Leibniz did– a quotient of a tiny change in f(x)f(x)f(x) and a tiny change in xxx. But this explanation comes with two important disclaimers. First, ddd is not a value. If it were a value, you could cancel out ddd’s in the numerator and the denominator. But you can’t. Instead think of ddd as an operator. When applied to f(x)f(x)f(x) or xxx, it produces an infinitesimally small quantity. Alternatively you can think of df(x)df(x)df(x) and dxdxdx as one symbol that h...