Section 6.4 Differentials

The tangent line to the graph of \(y=f(x)\) at the point (\(x_0\text{,}\)\(y_0\)) is given by
where the slope \(m\) is of course just the derivative \(\frac{df}{dx} \big|_{x=x_0}\text{.}\) It is tempting to rewrite the equation of the tangent line as
which is also used for linear approximation in the form
as shown in Figure 6.4.1. Regarding \(\Delta x\) as small, we rewrite (6.4.3) in the form
where the differential \(df\) can be interpreted as the corresponding small change in \(f\text{.}\)
The intuitive idea behind differentials is to consider the small quantities “\(dy\)” and “\(dx\)” separately, with the derivative \(\frac{dy}{dx}\) denoting their relative rate of change. So rather than either of the expressions used by Newton or Leibniz (see Section 6.3), if \(y=x^2\) we write
More generally, if \(y\) is any function of \(x\text{,}\) then the derivative \(\frac{dy}{dx}\) relates the differentials \(dy\) and \(dx\) via
You can safely think of (6.4.5) as a sufficiently small quantity, or as the numerator of Leibniz notation, or as shorthand for a limit argument, or in terms of differential forms, or nonstandard analysis, or ...; it doesn't matter, they all give the same equations.