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

76. The nth derivative. 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.

Illustrative Example 1. Given $y=e^{ax}$, find ${\frac {d^{n}y}{dx^{n}}}$.

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

Illustrative Example 2. Given $y=\ln x$, find ${\frac {d^{n}y}{dx^{n}}}$

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

Illustrative Example 3. Given $y=\sin x$, find ${\frac {d^{n}y}{dx^{n}}}$

Solution,	${\frac {dy}{dx}}=\cos x$	$=\sin \left(x+{\frac {\pi }{2}}\right)$,
	${\frac {d^{2}y}{dx^{2}}}={\frac {d}{dx}}\sin \left(x+{\frac {\pi }{2}}\right)$	$=\cos \left(x+{\frac {\pi }{2}}\right)=\sin \left(x+{\frac {2\pi }{2}}\right)$,
	${\frac {d^{3}y}{dx^{3}}}={\frac {d}{dx}}\sin \left(x+{\frac {2\pi }{2}}\right)$	$=\cos \left(x+{\frac {2\pi }{2}}\right)=\sin \left(x+{\frac {3\pi }{2}}\right)$
	.   .   .	.   .   .
∴	${\frac {d^{n}y}{dx^{n}}}$	$=\sin \left(x+{\frac {n\pi }{2}}\right)$. Ans.

77. Leibnitz's Formula for the nth derivative of a product. This formula expresses the nth derivative of the product of two variables in terms of the variables themselves and their successive derivatives.