The $n$-th derivative

For certain functions a general expression involving $n$ may be found for the $n$-th derivative. The usual plan is to find a number of the first successive derivatives, as many as may be necessary to discover their law of formation, and then by induction write down the $n$-th derivative.

Example 7.3.1   Given $y = e^{ax}$, find $\frac{d^n y}{dx^n}$.

Solution. $\frac{dy}{dx} = ae^{ax}$, $\frac{d^2 y}{dx^2} = a^2e^{ax}$, ..., $\frac{d^n y}{dx^n} = a^ne^{ax}$.

Example 7.3.2   Given $y = \log\, x$, find $\frac{d^n y}{dx^n}$.

Solution. $\frac{dy}{dx} = \frac{1}{x}$, $\frac{d^2 y}{dx^2} = -\frac{1}{x^2}$, $\frac{d^3 y}{dx^3} = \frac{1 \cdot 2}{x^3}$, $\frac{d^4 y}{dx^4} = \frac{1 \cdot 2 \cdot 3}{x^4}$, ... $\frac{d^n y}{dx^n} = (-1)^{n - 1} \frac{(n - 1)!}{x^n}$.

Example 7.3.3   Given $y = \sin\, x$, find $\frac{d^n y}{dx^n}$.

Solution. $\frac{dy}{dx} = \cos\, x = \sin \left ( x + \frac{\pi}{2} \right )$,

$$\frac{d^2 y}{dx^2} = \frac{d}{dx} \sin \left ( x + \frac{\pi}{2}... ...eft ( x + \frac{\pi}{2} \right ) = \sin \left ( x + \frac{2 \pi}{2} \right ),$$

$$\frac{d^3 y}{dx^3} = \frac{d}{dx} \sin \left ( x + \frac{2 \pi}{... ...ft ( x + \frac{2 \pi}{2} \right ) = \sin \left ( x + \frac{3 \pi}{2} \right )$$

$$\dots$$

$$\frac{d^n y}{dx^n} = \sin \left ( x + \frac{n \pi}{2} \right ).$$