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Section 6.3 Leibniz vs. Newton

Derivatives are instantaneous rates of change, which are in turn the ratios of small changes. There are two traditional notations for derivatives, which you have likely already seen.

Newton: In this notation, due to Newton, the primary objects are functions, such as \(f(x)=x^2\text{,}\) and derivatives are written with a prime, as in \(f'(x)=2x\text{.}\)  1 

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 \(\frac{dy}{dx}=2x\text{.}\)

Both notations are in common usage, and both 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!

Both of these notations distinguish between dependent quantities (\(f(x)\) or \(y\)) and the independent variable (\(x\)). However, in the real world one doesn't always know in advance which variables are independent. We therefore go one step further, and express derivatives in terms of differentials.

Newton actually used dots rather than primes, and used \(t\) rather than \(x\) as the independent variable. The use of primes and \(x\) is usually attributed to Lagrange, but was actually introduced by Euler. See also Section 9.2.