Successive differentials

As the differential of a function is in general also a function of the independent variable, we may deal with its differential. Consider the function

$$y = f(x).$$

$d(dy)$ is called the second differential of $y$ (or of the function) and is denoted by the symbol

$$d^2y.$$

Similarly, the third differential of $y$, $d[d(dy)]$, is written $d^3y$, and so on, to the $n$-th differential of $y$,

$$d^ny.$$

Since $dx$, the differential of the independent variable, is independent of $x$ (see §9.2), it must be treated as a constant when differentiating with respect to $x$. Bearing this in mind, we get very simple relations between successive differentials and successive derivatives.

For $dy = f'(x)dx$, and $d^2y = f''(x)(dx)^2$, since $dx$ is regarded as a constant. Also, $d^3y = f'''(x)(dx)^3$, and in general $d^ny = f^{(n)}(x)(dx)^n$.

Dividing both sides of each expression by the power of $dx$ occurring on the right, we get our ordinary derivative notation

$$\frac{d^2 y}{dx^2} = f''(x), \frac{d^3 y}{dx^3} = f'''(x), \dots, \frac{d^n y}{dx^n} = f^{(n)} (x).$$

Powers of an infinitesimal are called infinitesimals of a higher order. More generally, if for the infinitesimals $a$ and $b$, then $b$ is said to be an infinitesimal of a higher order than $a$.

Example 9.8.1   Find the third differential of $y = x^5 - 2x^3 + 3x - 5$.

Solution. $dy = (5x^4 - 6x^2 + 3)dx$, $d^2y = (20x^3 - 12x)(dx)^2$, $d^3y = (60x^2 - 12)(dx)^3$. .

NOTE. This is evidently the third derivative of the function multiplied by the cube of the differential of the independent variable. Dividing through by $(dx)^3$, we get the third derivative

$$\frac{d^3 y}{dx^3} = 60x^2 - 12.$$