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 d y d x dxdy. 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 ) h hf(x+h)−f(x)
when h h h is “infinitesimally small”. He denoted this infinitesimally small quantity of h h h by d x dx dx, and the corresponding difference f ( x + d x ) − f ( x ) f(x+dx)-f(x) f(x+dx)−f(x) by d f ( x ) df(x) df(x). Thus for a given function f f f the Leibniz notation for its derivative f ′ f’ f′ is:
d f ( x ) d x = f ′ dxdf(x)=f′
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