Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/122



For certain functions a general expression involving n may be found for the nth 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 nth derivative.

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




 * Solution,
 * style="text-align: right;"|$$\frac{dy}{dx}$$
 * $$= ae^{ax}$$,
 * style="text-align: right;"|$$\frac{d^2 y}{dx^2}$$
 * $$= a^2 e^{ax}$$
 * colspan="2"| .  ..
 * style="text-align: right;"|∴
 * style="text-align: right;"|$$\frac{d^n y}{dx^n}$$
 * $$= a^n e^{ax}$$. Ans.
 * }
 * style="text-align: right;"|∴
 * style="text-align: right;"|$$\frac{d^n y}{dx^n}$$
 * $$= a^n e^{ax}$$. Ans.
 * }
 * }

Given $$y = \ln x$$, find $$\frac{d^n y}{dx^n}$$




 * Solution,
 * style="text-align: right;"|$$\frac{dy}{dx}$$
 * $$= \frac{1}{x}$$,
 * style="text-align: right;"|$$\frac{d^2 y}{dx^2}$$
 * $$= -\frac{1}{x^2}$$
 * style="text-align: right;"|$$\frac{d^3 y}{dx^3}$$
 * $$= \frac{1 \cdot 2}{x^3}$$,
 * style="text-align: right;"|$$\frac{d^4 y}{dx^4}$$
 * $$= \frac{1 \cdot 2 \cdot 3}{x^4}$$,
 * colspan="2"| .  ..
 * style="text-align: right;"|∴
 * style="text-align: right;"|$$\frac{d^n y}{dx^n}$$
 * $$= (-1)^{n - 1} \frac{(n - 1)!}{x^n}$$. Ans.
 * }
 * $$= \frac{1 \cdot 2 \cdot 3}{x^4}$$,
 * colspan="2"| .  ..
 * style="text-align: right;"|∴
 * style="text-align: right;"|$$\frac{d^n y}{dx^n}$$
 * $$= (-1)^{n - 1} \frac{(n - 1)!}{x^n}$$. Ans.
 * }
 * style="text-align: right;"|$$\frac{d^n y}{dx^n}$$
 * $$= (-1)^{n - 1} \frac{(n - 1)!}{x^n}$$. Ans.
 * }

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




 * Solution,
 * style="text-align: right;"|$$\frac{dy}{dx} = \cos x$$
 * $$= \sin \left ( x + \frac{\pi}{2} \right )$$,
 * style="text-align: right;"|$$\frac{d^2 y}{dx^2} = \frac{d}{dx} \sin \left ( x + \frac{\pi}{2} \right )$$
 * $$= \cos \left ( x + \frac{\pi}{2} \right ) = \sin \left ( x + \frac{2 \pi}{2} \right )$$,
 * style="text-align: right;"|$$\frac{d^3 y}{dx^3} = \frac{d}{dx} \sin \left ( x + \frac{2 \pi}{2} \right )$$
 * $$= \cos \left ( x + \frac{2 \pi}{2} \right ) = \sin \left ( x + \frac{3 \pi}{2} \right )$$
 * style="text-align: right;"| .  ..
 * style="text-align: right;"|∴
 * style="text-align: right;"|$$\frac{d^n y}{dx^n}$$
 * $$= \sin \left ( x + \frac{n \pi}{2} \right )$$. Ans.
 * }
 * style="text-align: right;"| .  ..
 * style="text-align: right;"|∴
 * style="text-align: right;"|$$\frac{d^n y}{dx^n}$$
 * $$= \sin \left ( x + \frac{n \pi}{2} \right )$$. Ans.
 * }
 * style="text-align: right;"|$$\frac{d^n y}{dx^n}$$
 * $$= \sin \left ( x + \frac{n \pi}{2} \right )$$. Ans.
 * }

This formula expresses the nth derivative of the product of two variables in terms of the variables themselves and their successive derivatives.