Vipul: Created page with "{{differentiation rule}} ==Statement== For $k$ a positive integer, denote by $f^{(k)}$ the function obtained by differentiating $f$ a total of ..."

2011-09-05T14:03:08Z

Created page with "{{differentiation rule}} ==Statement== For <math>k</math> a positive integer, denote by <math>f^{(k)}</math> the function obtained by differentiating <math>f</math> a total of ..."

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{{differentiation rule}}

==Statement==

For <math>k</math> a positive integer, denote by <math>f^{(k)}</math> the function obtained by differentiating <math>f</math> a total of <math>k</math> times. The operation <math>f \mapsto f^{(k)}</math> is a linear operator. We give two equivalent ways of stating this below.

===In terms of additivity and pulling out scalars===

The following are true:

* ''Repeated differentiation is additive'', or ''<math>k^{th}</math> derivative of sum is sum of derivatives'': If <math>f</math> and <math>g</math> are functions that are both differentiable at <math>x = x_0</math>, we have:

<math>\frac{d^k}{dx^k}[f(x) + g(x)]_{x = x_0} = f^{(k)}(x_0) + g^{(k)}(x_0)</math>

or equivalently:

<math>(f + g)^{(k)}(x_0) = f^{(k)}(x_0) + g^{(k)}(x_0)</math>

In point-free notation:

<math>(f + g)^{(k)} = f^{(k)} + g^{(k)}</math>

* ''Constants (also called scalars) can be pulled out of differentiations'': If <math>f</math> is differentiable at <math>x = x_0</math> and <math>\lambda</math> is a real number, then:

<math>\frac{d^k}{dx^k}[\lambda f(x)]|_{x = x_0} = \lambda f^{(k)}(x_0)</math>

===In terms of generalized linearity===

Suppose <math>f_1, f_2, \dots, f_n</math> are functions that are all differentiable at a point <math>x_0</math> and <math>a_1, a_2, \dots, a_n</math> are real numbers. Then:

<math>\frac{d^k}{dx^k}[a_1f_1(x) + a_2f_2(x) + \dots + a_nf_n(x)]|_{x = x_0} = a_1f_1^{(k)}(x_0) + a_2f_2^{(k)}(x_0) + \dots + a_nf_n^{(k)}(x_0)</math>

==Related rules==

* [[Differentiation is linear]]
* [[Product rule for differentiation]]
* [[Product rule for higher derivatives]]
* [[Chain rule for differentiation]]
* [[Chain rule for higher derivatives]]
* [[Quotient rule for differentiation]]

Repeated differentiation is linear - Revision history

Vipul: Created page with "{{differentiation rule}} ==Statement== For k a positive integer, denote by f^{(k)} the function obtained by differentiating f a total of ..."

Vipul: Created page with "{{differentiation rule}} ==Statement== For $k$ a positive integer, denote by $f^{(k)}$ the function obtained by differentiating $f$ a total of ..."