## Section7.4Differentials

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

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

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

$$\Delta y = \frac{df}{dx} \Delta x\tag{7.4.2}$$

which is also used for linear approximation in the form

$$\Delta f = f(x+\Delta x)-f(x) \approx \frac{df}{dx} \Delta x\label{deltaf}\tag{7.4.3}$$

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

$$df = \frac{df}{dx} \,dx\tag{7.4.4}$$

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

$$dy = 2x\,dx .\label{zapex}\tag{7.4.5}$$

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

$$dy = \frac{dy}{dx} dx .\label{zapex2}\tag{7.4.6}$$

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.