Definitions

If $f'(x)$ is the derivative of $f(x)$ for a particular value of $x$, and $\Delta x$ is an arbitrarily chosen^9.1 increment of $x$, then the differential of $f(x)$, denoted by the symbol $df(x)$, is defined by the equation

$$df(x) = f'(x)\Delta x.$$ (9.1)

If now $f(x) = x$, then $f'(x) = 1$, and (9.1) reduces to $dx = \Delta x$, showing that when $x$ is the independent variable, the differential of $x$ ($= dx$) is identical with $\Delta x$. Hence, if $y = f(x)$, (9.1) may in general be written in the form

$$dy = f'(x)\, dx.$$ (9.2)

The differential of a function equals its derivative multiplied by the differential of the independent variable.

On account of the position which the derivative $f'(x)$ here occupies, it is sometimes called the differential coefficient. The student should observe the important fact that, since $dx$ may be given any arbitrary value whatever, $dx$ is independent of $x$. Hence, $dy$ is a function of two independent variables $x$ and $dx$.

Let us illustrate what this means geometrically.

Figure 9.1: The differential of a function.

Let $f'(x)$ be the derivative of $y = f(x)$ at P. Take $dx = PQ$, then

$$dy = f'(x)dx = \tan \tau \cdot PQ = \frac{QT}{PQ} \cdot PQ = QT.$$

Therefore $dy$, or $df(x)$, is the increment ($= QT$) of the ordinate of the tangent corresponding^9.2 to $dx$.

This gives the following interpretation of the derivative as a fraction.

If an arbitrarily chosen increment of the independent variable $x$ for a point $(x, y)$ on the curve $y = f(x)$ be denoted by $dx$, then in the derivative

$$\frac{dy}{dx} = f'(x) = \tan \tau,$$

$dy$ denotes the corresponding increment of the ordinate drawn to the tangent.