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Section 7.4 Differentials

Figure 7.4.1. Linear approximation using the tangent line.

The tangent line to the graph of \(y=f(x)\) at the point (\(x_0\text{,}\)\(y_0\)) is given by

\begin{equation} y-y_0 = m \left( x-x_0 \right)\tag{7.4.1} \end{equation}

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

\begin{equation} \Delta y = \frac{df}{dx} \Delta x\tag{7.4.2} \end{equation}

which is also used for linear approximation in the form

\begin{equation} \Delta f = f(x+\Delta x)-f(x) \approx \frac{df}{dx} \Delta x\tag{7.4.3} \end{equation}

as shown in Figure 7.4.1. Regarding \(\Delta x\) as small, we rewrite (7.4.3) in the form

\begin{equation} df = \frac{df}{dx} \,dx\tag{7.4.4} \end{equation}

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 7.3), if \(y=x^2\) we write

\begin{equation} dy = 2x\,dx .\tag{7.4.5} \end{equation}

More generally, if \(y\) is any function of \(x\text{,}\) then the derivative \(\frac{dy}{dx}\) relates the differentials \(dy\) and \(dx\) via

\begin{equation} dy = \frac{dy}{dx} dx .\tag{7.4.6} \end{equation}

You can safely think of (7.4.5) 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.