Higher derivative - Revision history

Vipul: /* Leibniz notation */

2011-10-16T16:41:45Z

Leibniz notation

← Older revision		Revision as of 16:41, 16 October 2011
Line 43:		Line 43:
	and is defined as:		and is defined as:

	<math>\! \frac{d^ky{dx^k} = \frac{d}{dx}\left[\frac{d}{dx} \left[ \dots \left[\frac{d}{dx}(y)\right] \dots \right] \right]</math>		<math>\! \frac{d^ky}{dx^k} = \frac{d}{dx}\left[\frac{d}{dx} \left[ \dots \left[\frac{d}{dx}(y)\right] \dots \right] \right]</math>

	where the <math>d/dx</math> occurs <math>k</math> times. Alternatively we can define it inductively as:		where the <math>d/dx</math> occurs <math>k</math> times. Alternatively we can define it inductively as:

Vipul: /* Function and prime notation */

2011-10-16T16:41:10Z

Function and prime notation

← Older revision		Revision as of 16:41, 16 October 2011
Line 7:		Line 7:
	===Function and prime notation===		===Function and prime notation===

	Suppose <math>f</math> is a [[function]] and <math>k</math> is a nonnegative integer. The <math>k^{th}</math> derivative of <math>f</math>, denoted <math>f^{(k)}<math> or <math>f^{'''\dots'}</math> where <math>{}^'</math> occurs a total of <math>k</math> times, is defined as the function obtained by differentiating <math>f</math> a total of <math>k</math> times (i.e., taking the derivative, then taking the derivative of that, and so on, a total of <math>k</math> times). The first few cases are shown explicitly:		Suppose <math>f</math> is a [[function]] and <math>k</math> is a nonnegative integer. The <math>k^{th}</math> derivative of <math>f</math>, denoted <math>f^{(k)}</math> or <math>f^{'''\dots'}</math> where <math>{}^'</math> occurs a total of <math>k</math> times, is defined as the function obtained by differentiating <math>f</math> a total of <math>k</math> times (i.e., taking the derivative, then taking the derivative of that, and so on, a total of <math>k</math> times). The first few cases are shown explicitly:

	{\| class="sortable" border="1"		{\| class="sortable" border="1"

Vipul: Created page with "==Definition== ===Terminology=== '''Higher derivatives''' are also called '''repeated derivatives''' or '''iterated derivatives'''. ===Function and prime notation=== Suppose ..."

2011-10-16T16:40:36Z

Created page with "==Definition== ===Terminology=== '''Higher derivatives''' are also called '''repeated derivatives''' or '''iterated derivatives'''. ===Function and prime notation=== Suppose ..."

New page

==Definition==

===Terminology===

'''Higher derivatives''' are also called '''repeated derivatives''' or '''iterated derivatives'''.

===Function and prime notation===

Suppose <math>f</math> is a [[function]] and <math>k</math> is a nonnegative integer. The <math>k^{th}</math> derivative of <math>f</math>, denoted <math>f^{(k)}<math> or <math>f^{'''\dots'}</math> where <math>{}^'</math> occurs a total of <math>k</math> times, is defined as the function obtained by differentiating <math>f</math> a total of <math>k</math> times (i.e., taking the derivative, then taking the derivative of that, and so on, a total of <math>k</math> times). The first few cases are shown explicitly:

{| class="sortable" border="1"
! Value of <math>k</math> !! Notation with repeated primes for <math>f^{(k)}</math> !! <matH>f^{(k)}</math> notation !! Definition !! In words
|-
| 0 || <math>\! f</math> || <math>\! f^{(0)}</math> || <math>\! f</math> || the original function
|-
| 1 || <math>\! f'</math> || <math>\! f^{(1)}</math> || <math>\! f'</math> || the [[derivative]], also called the '''first derivative'''
|-
| 2 || <math>\! f''</math> || <math>\! f^{(2)}</math> || <math>\! (f')^'</math> || the [[second derivative]]
|-
| 3 || <math>\! f'''</math> || <matH>\! f^{(3)}</math> || <math>\! ((f')^')^'</math> || the [[third derivative]]
|}

We could also define the <math>k^{th}</math> derivative inductively as:

<math>\! f^{(k)} = (f^{(k-1)})^'</math>

or as:

<math>\! f^{(k)} = (f')^{(k-1)}</math>

with the base case <math>f^{(0)} = f</math>.

===Leibniz notation===

Suppose <math>y = f(x)</math>, so <math>y</math> is a ''dependent variable'' depending on <math>x</math>, the ''independent variable''. The <math>k^{th}</math> derivative of <math>y</math> with respect to <math>x</math> is denoted:

<math>\! \frac{d^k}{(dx)^k}y</math>

or as:

<math>\! \frac{d^ky}{dx^k}</math>

and is defined as:

<math>\! \frac{d^ky{dx^k} = \frac{d}{dx}\left[\frac{d}{dx} \left[ \dots \left[\frac{d}{dx}(y)\right] \dots \right] \right]</math>

where the <math>d/dx</math> occurs <math>k</math> times. Alternatively we can define it inductively as:

<math>\! \frac{d^ky}{dx^k} = \frac{d}{dx} \frac{d^{k-1}y}{dx^{k-1}}</math>

with the base case <math>k = 0</math> being defined as <math>\frac{d^0y}{dx^0} = y</math>.