Chain rule for partial differentiation - Revision history

Vipul at 04:29, 27 January 2013

2013-01-27T04:29:44Z

← Older revision		Revision as of 04:29, 27 January 2013
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	<center>{{#widget:YouTube\|id=~~-bBNTxI37Ng~~}}</center>		<center>{{#widget:YouTube\|id=I98HWN2sbLs}}</center>

	===Conceptual statement for a two-step composition===		===Conceptual statement for a two-step composition===
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	As noted above, in those cases where the functions involved have only one input, the partial derivative becomes an ordinary [[derivative]].		As noted above, in those cases where the functions involved have only one input, the partial derivative becomes an ordinary [[derivative]].

	<center>{{#widget:YouTube\|id=~~nnapNypMgkk~~}}</center>		<center>{{#widget:YouTube\|id=YBXQWnu_F_I}}</center>

	===Statement with symbols for a two-step composition===		===Statement with symbols for a two-step composition===
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	<math>\frac{\partial w_i}{\partial x_j} = \sum_{k=1}^m \frac{\partial w_i}{\partial u_k} \frac{\partial u_k}{\partial x_j}</math>		<math>\frac{\partial w_i}{\partial x_j} = \sum_{k=1}^m \frac{\partial w_i}{\partial u_k} \frac{\partial u_k}{\partial x_j}</math>

	<center>{{#widget:YouTube\|id=~~f3Ko3ste0_o~~}}</center>		<center>{{#widget:YouTube\|id=Ij7vxWAZP-M}}</center>

	==Related facts==		==Related facts==

Vipul at 19:22, 8 April 2012

2012-04-08T19:22:23Z

← Older revision		Revision as of 19:22, 8 April 2012
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	As noted above, in those cases where the functions involved have only one input, the partial derivative becomes an ordinary [[derivative]].		As noted above, in those cases where the functions involved have only one input, the partial derivative becomes an ordinary [[derivative]].

	{{#widget:YouTube\|id=nnapNypMgkk}}		<center>{{#widget:YouTube\|id=nnapNypMgkk}}</center>

	===Statement with symbols for a two-step composition===		===Statement with symbols for a two-step composition===
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	<math>\frac{\partial w_i}{\partial x_j} = \sum_{k=1}^m \frac{\partial w_i}{\partial u_k} \frac{\partial u_k}{\partial x_j}</math>		<math>\frac{\partial w_i}{\partial x_j} = \sum_{k=1}^m \frac{\partial w_i}{\partial u_k} \frac{\partial u_k}{\partial x_j}</math>

			<center>{{#widget:YouTube\|id=f3Ko3ste0_o}}</center>

	==Related facts==		==Related facts==

Vipul: /* Conceptual statement for a two-step composition */

2012-04-08T19:21:34Z

Conceptual statement for a two-step composition

← Older revision		Revision as of 19:21, 8 April 2012
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	As noted above, in those cases where the functions involved have only one input, the partial derivative becomes an ordinary [[derivative]].		As noted above, in those cases where the functions involved have only one input, the partial derivative becomes an ordinary [[derivative]].

			{{#widget:YouTube\|id=nnapNypMgkk}}

	===Statement with symbols for a two-step composition===		===Statement with symbols for a two-step composition===

Vipul at 19:19, 8 April 2012

2012-04-08T19:19:37Z

← Older revision		Revision as of 19:19, 8 April 2012
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	\|}		\|}

	{{#widget:YouTube\|id=-bBNTxI37Ng}}		<center>{{#widget:YouTube\|id=-bBNTxI37Ng}}</center>

	===Conceptual statement for a two-step composition===		===Conceptual statement for a two-step composition===

Vipul at 19:19, 8 April 2012

2012-04-08T19:19:12Z

← Older revision		Revision as of 19:19, 8 April 2012
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	\|}		\|}

			{{#widget:YouTube\|id=-bBNTxI37Ng}}

	===Conceptual statement for a two-step composition===		===Conceptual statement for a two-step composition===

Vipul: /* Conceptual statement for a two-step composition */

2012-04-08T19:17:38Z

Conceptual statement for a two-step composition

← Older revision		Revision as of 19:17, 8 April 2012
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	<math>\mbox{Partial derivative of any dependent output variable with respect to any independent input variable} </math>		<math>\mbox{Partial derivative of any dependent output variable with respect to any independent input variable} </math>

	<math>= \sum_{\mbox{all intermediate variables}} \mbox{Partial derivative of output variable with respect to input variable via intermediate variable}</math>		<math>= \sum_{\mbox{all intermediate variables}} \mbox{(Partial derivative of output variable with respect to input variable via intermediate variable)}</math>

	<math> = \sum_{\mbox{all intermediate variables}} \frac{\partial(\mbox{output variable})}{\partial(\mbox{intermediate variable})} \frac{\partial(\mbox{intermediate variable})}{\partial(\mbox{input variable})}</math>		<math> = \sum_{\mbox{all intermediate variables}} \frac{\partial(\mbox{output variable})}{\partial(\mbox{intermediate variable})} \frac{\partial(\mbox{intermediate variable})}{\partial(\mbox{input variable})}</math>

Vipul: /* Conceptual statement for a two-step composition */

2012-04-08T19:17:06Z

Conceptual statement for a two-step composition

← Older revision		Revision as of 19:17, 8 April 2012
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	<math>= \sum_{\mbox{all intermediate variables}} \mbox{Partial derivative of output variable with respect to input variable via intermediate variable}</math>		<math>= \sum_{\mbox{all intermediate variables}} \mbox{Partial derivative of output variable with respect to input variable via intermediate variable}</math>

	<math> = \sum_{\mbox{all intermediate variables}} \mbox{~~(Partial derivative of~~ output variable ~~with respect to~~ intermediate variable) ~~times~~ (~~Partial derivative of~~ intermediate variable ~~with respect to~~ input variable)}</math>		<math> = \sum_{\mbox{all intermediate variables}} \frac{\partial(\mbox{output variable})}{\partial(\mbox{intermediate variable})} \frac{\partial(\mbox{intermediate variable})}{\partial(\mbox{input variable})}</math>

	In other words, we get in general a sum of products, each product being of two partial derivatives involving the intermediate variable.		In other words, we get in general a sum of products, each product being of two partial derivatives involving the intermediate variable.

Vipul: /* Conceptual statement for a two-step composition */

2012-04-08T19:15:49Z

Conceptual statement for a two-step composition

← Older revision		Revision as of 19:15, 8 April 2012
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	Then, we have:		Then, we have:

	<math>\mbox{Partial derivative of any dependent output variable with respect to any independent input ~~variable} = \sum_{\mbox{all intermediate variables}} \mbox{Partial derivative of output variable with respect to input variable via intermediate~~ variable}</math>		<math>\mbox{Partial derivative of any dependent output variable with respect to any independent input variable} </math>

	~~More explicitly:~~		<math>= \sum_{\mbox{all intermediate variables}} \mbox{Partial derivative of output variable with respect to input variable via intermediate variable}</math>

	<math>~~\mbox{Partial derivative of any dependent output variable with respect to any independent input variable}~~ = \sum_{\mbox{all intermediate variables}} \mbox{(Partial derivative of output variable with respect to intermediate variable) times (Partial derivative of intermediate variable with respect to input variable)}</math>		<math> = \sum_{\mbox{all intermediate variables}} \mbox{(Partial derivative of output variable with respect to intermediate variable) times (Partial derivative of intermediate variable with respect to input variable)}</math>

	In other words, we get in general a sum of products, each product being of two partial derivatives involving the intermediate variable.		In other words, we get in general a sum of products, each product being of two partial derivatives involving the intermediate variable.

Vipul at 19:14, 8 April 2012

2012-04-08T19:14:57Z

← Older revision		Revision as of 19:14, 8 April 2012
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	Then, we have:		Then, we have:

	Partial derivative of any dependent output variable with respect to any independent input variable = ~~<math>~~\sum_{\mbox{all intermediate variables}}~~</math>~~ Partial derivative of output variable with respect to input variable ''via'' intermediate variable = <math>\sum_{\mbox{all intermediate variables}}~~</math>~~ (Partial derivative of output variable with respect to intermediate variable) times (Partial derivative of intermediate variable with respect to input variable)		<math>\mbox{Partial derivative of any dependent output variable with respect to any independent input variable} = \sum_{\mbox{all intermediate variables}} \mbox{Partial derivative of output variable with respect to input variable via intermediate variable}</math>

			More explicitly:

			<math>\mbox{Partial derivative of any dependent output variable with respect to any independent input variable} = \sum_{\mbox{all intermediate variables}} \mbox{(Partial derivative of output variable with respect to intermediate variable) times (Partial derivative of intermediate variable with respect to input variable)}</math>

	In other words, we get in general a sum of products, each product being of two partial derivatives involving the intermediate variable.		In other words, we get in general a sum of products, each product being of two partial derivatives involving the intermediate variable.

Vipul: Created page with "==Statement== The statement explains how to differentiate composites involving functions of more than one variable, where differentiate is in the sense of computing [[partial..."

2012-04-07T23:52:59Z

Created page with "==Statement== The statement explains how to differentiate composites involving functions of more than one variable, where differentiate is in the sense of computing [[partial..."

New page

==Statement==

The statement explains how to differentiate composites involving functions of more than one variable, where differentiate is in the sense of computing [[partial derivative]]s. Note that in those cases where the functions involved have only one input, the partial derivative becomes an ordinary [[derivative]].

===Statement for function of two variables composed with two functions of one variable===

{| class="sortable" border="1"
! Version type !! Statement
|-
| Mixed functional, dependent variable notation (generic point) || Suppose <math>f,g</math> are both functions of one variable and <math>h</math> is a function of two variables. Suppose <math>u = f(x), v = g(x), w = h(u,v)</math>. Then:<br><math>\frac{dw}{dx} = \frac{\partial w}{\partial u}\frac{du}{dx} + \frac{\partial w}{\partial v} \frac{dv}{dx} = h_u(u,v)f'(x) + h_v(u,v)g'(x)</math>
|-
| Pure functional notation (generic point) || Suppose <math>f,g</math> are both functions of one variable and <math>h</math> is a function of two variables. Consider the function:<br><math>H(x) := h(f(x),g(x))</math><br>Then, we have <math>H'(x) = h_1(f(x),g(x))f'(x) + h_2(f(x),g(x))g'(x)</math> where <math>h_1, h_2</math> denote respectively the partial derivatives with respect to the first and second coordinates.
|-
| Pure dependent variable notation (generic point)|| Suppose <math>u,v</math> are variables functionally dependent on <math>x</math> and <math>w</math> is a variable functionally dependent on both <math>u</math> and <math>v</math>. Then, <math>\frac{dw}{dx} = \frac{\partial w}{\partial u}\frac{du}{dx} + \frac{\partial w}{\partial v} \frac{dv}{dx}</math>
|}

===Conceptual statement for a two-step composition===

Consider a situation where we have three kinds of variables:

* ''Independent input variables''
* ''Dependent intermediate variables'', each of which is a function of the input variables.
* ''Dependent output variables'', each of which is a function of the intermediate variables.

Then, we have:

Partial derivative of any dependent output variable with respect to any independent input variable = <math>\sum_{\mbox{all intermediate variables}}</math> Partial derivative of output variable with respect to input variable ''via'' intermediate variable = <math>\sum_{\mbox{all intermediate variables}}</math> (Partial derivative of output variable with respect to intermediate variable) times (Partial derivative of intermediate variable with respect to input variable)

In other words, we get in general a sum of products, each product being of two partial derivatives involving the intermediate variable.

As noted above, in those cases where the functions involved have only one input, the partial derivative becomes an ordinary [[derivative]].

===Statement with symbols for a two-step composition===

Suppose we have:

* ''Independent input variables'' <math>x_1,x_2,\dots,x_n</math>
* ''Dependent intermediate variables'', <math>u_1,u_2,\dots,u_m</math>, each of which is a function of <math>x_1,x_2,\dots,x_n</math>.
* ''Dependent output variables'' <math>w_1,w_2,\dots,w_p</math>, each of which of a function of <math>u_1,u_2,\dots,u_m</math>.

Then, for any <math>i \in \{ 1,2,3,\dots,p \}</math> and <math>j \in \{ 1,2,3,\dots,n\}</math>, we have:

<math>\frac{\partial w_i}{\partial x_j} = \sum_{k=1}^m \frac{\partial w_i}{\partial u_k} \frac{\partial u_k}{\partial x_j}</math>

==Related facts==

===Applications===

* [[Product rule for differentiation]]: See [[proof of product rule for differentiation using chain rule for partial differentiation]]