Partial derivative - Revision history

Vipul: /* For a function of two variables */

2013-01-25T16:58:55Z

For a function of two variables

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	===For a function of multiple variables===		===For a function of multiple variables===

2013-01-25T03:06:25Z

Graphical interpretation

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	===For a function of multiple variables at a point===		===For a function of multiple variables at a point===

2013-01-25T03:03:55Z

For a function of two variables

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	===For a function of multiple variables===		===For a function of multiple variables===

2013-01-25T02:59:06Z

Generic definition

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	Suppose <math>f</math> is a function of more than one variable, where <math>x</math> is one of the input variables to <math>f</math>. Fix a choice <math>x = x_0</math> and fix the values of all the other variables. The '''partial derivative''' of <math>f</math> with respect to <math>x</math> ''at the point'', denoted <math>\partial f/\partial x</math>, or <math>f_x</math>, is defined as the derivative at <math>x_0</math> of the function that sends <math>x</math> to <math>f</math> at <math>x</math> for the same fixed choice of the other input variables.		Suppose <math>f</math> is a function of more than one variable, where <math>x</math> is one of the input variables to <math>f</math>. Fix a choice <math>x = x_0</math> and fix the values of all the other variables. The '''partial derivative''' of <math>f</math> with respect to <math>x</math> ''at the point'', denoted <math>\partial f/\partial x</math>, or <math>f_x</math>, is defined as the derivative at <math>x_0</math> of the function that sends <math>x</math> to <math>f</math> at <math>x</math> for the same fixed choice of the other input variables.

	<center>{{#widget:YouTube\|id=~~wPEMBebMm64~~}}</center>		<center>{{#widget:YouTube\|id=rsZc1SYuiig}}</center>

	===For a function of two variables===		===For a function of two variables===

2013-01-25T02:51:39Z

For a function of multiple variables

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	==Definition as a function==		==Definition as a function==

2013-01-25T02:46:07Z

For a function of two variables

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	===For a function of multiple variables===		===For a function of multiple variables===

2013-01-25T02:32:57Z

For a function of two variables

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	Note that the above only refers to the points at which it makes sense to ''try'' computing the partial derivative. It may still turn out that the partial derivative does not exist at many of these points.		Note that the above only refers to the points at which it makes sense to ''try'' computing the partial derivative. It may still turn out that the partial derivative does not exist at many of these points.

	<center>{{#widget:YouTube\|id=~~oCzLzPCUPCU~~}}</center>		<center>{{#widget:YouTube\|id=9Q9OrXye748}}</center>

	==Caveats==		==Caveats==

2012-05-13T02:38:47Z

Graphical interpretation

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	\| The partial derivative <math>f_y(x_0,y_0)</math> at a point <math>(x_0,y_0)</math> in the domain of the function \|\| The slope of the tangent line at <math>y = y_0</math> to the restriction of the graph of <math>f</math> to the plane <math>x = x_0</math>.		\| The partial derivative <math>f_y(x_0,y_0)</math> at a point <math>(x_0,y_0)</math> in the domain of the function \|\| The slope of the tangent line at <math>y = y_0</math> to the restriction of the graph of <math>f</math> to the plane <math>x = x_0</math>.
	\|}		\|}

			<center>{{#widget:YouTube\|id=7AP2uSrJTcA}}</center>

	===For a function of multiple variables at a point===		===For a function of multiple variables at a point===

2012-05-08T16:07:13Z

Graphical interpretation

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 | The partial derivative <math>f_y(x_0,y_0)</math> at a point <math>(x_0,y_0)</math> in the domain of the function || The slope of the tangent line at <math>y = y_0</math> to the restriction of the graph of <math>f</math> to the plane <math>x = x_0</math>.
 |}
 ==Related notions==

2012-05-08T15:56:10Z

Related notions

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 |-
 | Definition as a [[directional derivative]] || Directional derivative in the positive <math>x_i</math>-direction.
 |}