Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/121

SUCCESSIVE DIFFERENTIATION

74. Definition of successive derivatives. We have seen that the derivative of a function of x is in general also a function of x. This new function may also be differentiable, in which case the derivative of the first derivative is called the second derivative of the original function. Similarly, the derivative of the second derivative is called the third derivative; and so on to the nth derivative. Thus, if

$\ y$	$=\ 3x^{4}$
${\frac {dy}{dx}}$	$=\ 12x^{3}$
${\frac {d}{dx}}\left({\frac {dy}{dx}}\right)$	$=\ 36x^{2}$
${\frac {d}{dx}}\left[{\frac {d}{dx}}\left({\frac {dy}{dx}}\right)\right]$	$=\ 72x$, etc.

75. Notation. The symbols for the successive derivatives are usually abbreviated as follows:

${\frac {d}{dx}}\left({\frac {dy}{dx}}\right)$	$=\ {\frac {d^{2}y}{dx^{2}}}$
${\frac {d}{dx}}\left[{\frac {d}{dx}}\left({\frac {dy}{dx}}\right)\right]$	$=\ {\frac {d}{dx}}\left({\frac {d^{2}y}{dx^{2}}}\right)={\frac {d^{3}y}{dx^{3}}}$,
. . .	. . .
${\frac {d}{dx}}\left({\frac {d^{n-1}y}{dx^{n-1}}}\right)$	$={\frac {d^{n}y}{dx^{n}}}$.

If $y=f(x)$, the successive derivatives are also denoted by

$f'(x),\ f''(x),\ f'''(x),\ f^{iv}(x),\ ...,\ f^{(n)}(x)$;

$y',\ y'',\ y''',\ y^{iv},\ ...,\ y^{(n)}$;

or,

${\frac {d}{dx}}f(x),\ {\frac {d^{2}}{dx^{2}}}f(x),\ {\frac {d^{3}}{dx^{3}}}f(x),\ {\frac {d^{4}}{dx^{4}}}f(x)\ ...,\ {\frac {d^{n}}{dx^{n}}}f(x)$.