Derivative Notation

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Section 5.2 Derivative Notation

There are two traditional notations for derivatives, which you have likely already seen.

Newton/Lagrange/Euler: In this notation, the primary objects are functions, such as \(f(x)=x^2\text{,}\) and derivatives are written with a prime, as in \(f'(x)=2x\text{.}\)

This notation is often referred to as “Newtonian”, but Newton actually used dots rather than primes, and used \(t\) rather than \(x\) as the independent variable. The use of primes and \(x\) is often attributed to Lagrange, but was in fact introduced by Euler.

Leibniz: In this notation, due to Leibniz, the primary objects are relationships, such as \(y=x^2\text{,}\) and derivatives are written as a ratio, as in \(\frac{dy}{dx}=2x\text{.}\)

These notations extend naturally to higher derivatives. We summarize the disscussion so far as follows:

Notation 5.2.1. Derivatives.

There are several different notations for derivatives in common use. You should be comfortable with all of them. Leibniz's notation for derivatives is:

\begin{equation} \frac{dy}{dx}, \qquad \frac{d^2 y}{dx^2}, \qquad \frac{d^3 y}{dx^3}, \qquad\dots\qquad \frac{d^n y}{dx^n}\tag{5.2.1} \end{equation}

Lagrange's notation for the same derivatives is: ¹

\begin{equation} y^{\prime}, \qquad y^{\prime\prime},\qquad y^{\prime\prime\prime}, \qquad \dots \qquad y^{(n)}\tag{5.2.2} \end{equation}

It is also common to set \(y=f(x)\) and write \(f'(x)\) instead of \(y'\text{,}\) etc.

Newton used dots instead of primes (and \(t\) as the independent variable, rather than \(x\)):

\begin{equation} \dot{y}, \qquad \ddot{y}, \qquad \dots \qquad y^{(n)}\tag{5.2.3} \end{equation}

All of these notations work fine for functions of a single variable. However, Leibniz notation is better suited to situations involving many quantities that are changing, both because it keeps explicit track of which derivative you took (“with respect to \(x\)”), and because it emphasizes that derivatives are ratios. Among other things, this helps you get the units right; mph are a ratio of miles to hours!

This notation was originally introduced by Euler.