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Section 9.2 Derivative Notation

Notation 9.2. Derivatives.

As pointed out in Section 6.3, 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{9.2.1} \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{9.2.2} \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{9.2.3} \end{equation}
This notation was actually introduced by Euler.