## Section6.2Derivative 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:

### Notation6.1.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:

$$\frac{dy}{dx}, \qquad \frac{d^2 y}{dx^2}, \qquad \frac{d^3 y}{dx^3}, \qquad\dots\qquad \frac{d^n y}{dx^n}\tag{6.2.1}$$

Lagrange's notation for the same derivatives is:  1

$$y^{\prime}, \qquad y^{\prime\prime},\qquad y^{\prime\prime\prime}, \qquad \dots \qquad y^{(n)}\tag{6.2.2}$$

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 $$x$$):

$$\dot{y}, \qquad \ddot{y}, \qquad \dots \qquad y^{(n)}\tag{6.2.3}$$

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.