Difference between revisions of "Product rule for partial differentiation"

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(Statement for two functions)
 
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==Statement for two functions==
 
==Statement for two functions==
  
===Statement for partial derivatives===
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===Statement for partial derivatives for functions of two variables===
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The derivatives used here are [[partial derivative]]s.
  
 
{| class="sortable" border="1"
 
{| class="sortable" border="1"
! Version type !! Statement for functions of two variables
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! Version type !! Statement  
 
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|-
 
| specific point, named functions || Suppose <math>f,g</math> are both functions of variables <math>x,y</math>. Suppose <math>(x_0,y_0)</math> is a point in the domain of both <math>f</math> and <math>g</math>. Suppose the partial derivatives <math>f_x(x_0,y_0)</math> and <math>g_x(x_0,y_0)</math> both exist. Let <math>fg</math> denote the [[pointwise product of functions|product]] of the functions. Then, we have:<br><math>(fg)_x(x_0,y_0) =f_x(x_0,y_0)g(x_0,y_0) + f(x_0,y_0)g_x(x_0,y_0)</math><br>Suppose the partial derivatives <math>f_y(x_0,y_0)</math> and <math>g_y(x_0,y_0)</math> both exist. Then, we have:<br><math>(fg)_y(x_0,y_0) = f_y(x_0,y_0)g(x_0,y_0) + f(x_0,y_0)g_y(x_0,y_0)</math>
 
| specific point, named functions || Suppose <math>f,g</math> are both functions of variables <math>x,y</math>. Suppose <math>(x_0,y_0)</math> is a point in the domain of both <math>f</math> and <math>g</math>. Suppose the partial derivatives <math>f_x(x_0,y_0)</math> and <math>g_x(x_0,y_0)</math> both exist. Let <math>fg</math> denote the [[pointwise product of functions|product]] of the functions. Then, we have:<br><math>(fg)_x(x_0,y_0) =f_x(x_0,y_0)g(x_0,y_0) + f(x_0,y_0)g_x(x_0,y_0)</math><br>Suppose the partial derivatives <math>f_y(x_0,y_0)</math> and <math>g_y(x_0,y_0)</math> both exist. Then, we have:<br><math>(fg)_y(x_0,y_0) = f_y(x_0,y_0)g(x_0,y_0) + f(x_0,y_0)g_y(x_0,y_0)</math>
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|}
  
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<center>{{#widget:YouTube|id=lTFCy8V5qDc}}</center>
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===Statement for partial derivatives for functions of multiple variables===
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{| class="sortable" border="1"
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! Version type !! Statement
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|-
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| specific point, named functions || Suppose <math>f,g</math> are both functions of variables <math>x_1,x_2,\dots,x_n</math>. Suppose <math>(a_1,a_2,\dots,a_n)</math> is a point in the domain of both <math>f</math> and <math>g</math>. Fix a number <math>i</math> in <math>\{ 1,2,3,\dots,n \}</math>. Suppose the partial derivatives <math>f_{x_i}(a_1,a_2,\dots,a_n)</math> and <math>g_{x_i}(a_1,a_2,\dots,a_n)</math> both exist. Let <math>fg</math> denote the [[pointwise product of functions|product]] of the functions. Then, we have:<br><math>(fg)_{x_i}(a_1,a_2,\dots,a_n) =f_{x_i}(a_1,a_2,\dots,a_n)g(a_1,a_2,\dots,a_n) + f(a_1,a_2,\dots,a_n)g_{x_i}(a_1,a_2,\dots,a_n)</math>
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|-
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| generic point, named functions || Suppose <math>f,g</math> are both functions of variables <math>x_1,_2,\dots,x_n</math>. Then, for any fixed <math>i</math> in <math>\{ 1,2,3,\dots,n \}</math>: <math>(fg)_{x_i}(x_1,x_2,\dots,x_n) =f_{x_i}(x_1,x_2,\dots,x_n)g(x_1,x_2,\dots,x_n) + f(x_1,x_2,\dots,x_n)g_{x_i}(x_1,x_2,\dots,x_n)</math><br>These hold wherever the right side expressions make sense (see [[concept of equality conditional to existence of one side]]).
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| generic point, named functions, point-free notation || Suppose <math>f,g</math> are both functions of variables <math>x,y</math>. Then, for any fixed <math>i</math> in <math>\{ 1,2,3,\dots,n \}</math>: <math>(fg)_{x_i} = f_{x_i}g + fg_{x_i}</math><br>These hold wherever the right side expressions make sense (see [[concept of equality conditional to existence of one side]]).
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===Statement for directional derivatives===
 
===Statement for directional derivatives===
  
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The rule applies at all points where the right side make sense.
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===Statement for gradient vectors===
 
===Statement for gradient vectors===

Latest revision as of 21:38, 8 April 2012

Statement for two functions

Statement for partial derivatives for functions of two variables

The derivatives used here are partial derivatives.

Version type Statement
specific point, named functions Suppose f,g are both functions of variables x,y. Suppose (x_0,y_0) is a point in the domain of both f and g. Suppose the partial derivatives f_x(x_0,y_0) and g_x(x_0,y_0) both exist. Let fg denote the product of the functions. Then, we have:
(fg)_x(x_0,y_0) =f_x(x_0,y_0)g(x_0,y_0) + f(x_0,y_0)g_x(x_0,y_0)
Suppose the partial derivatives f_y(x_0,y_0) and g_y(x_0,y_0) both exist. Then, we have:
(fg)_y(x_0,y_0) = f_y(x_0,y_0)g(x_0,y_0) + f(x_0,y_0)g_y(x_0,y_0)
generic point, named functions Suppose f,g are both functions of variables x,y.
(fg)_x(x,y) =f_x(x,y)g(x,y) + f(x,y)g_x(x,y)
(fg)_y(x,y) = f_y(x,y)g(x,y) + f(x,y)g_y(x,y)
These hold wherever the right side expressions make sense (see concept of equality conditional to existence of one side).
generic point, named functions, point-free notation Suppose f,g are both functions of variables x,y.
(f g)_x =f_xg + fg_x
(f g)_y = f_yg + fg_y
These hold wherever the right side expressions make sense (see concept of equality conditional to existence of one side).

Statement for partial derivatives for functions of multiple variables

Version type Statement
specific point, named functions Suppose f,g are both functions of variables x_1,x_2,\dots,x_n. Suppose (a_1,a_2,\dots,a_n) is a point in the domain of both f and g. Fix a number i in \{ 1,2,3,\dots,n \}. Suppose the partial derivatives f_{x_i}(a_1,a_2,\dots,a_n) and g_{x_i}(a_1,a_2,\dots,a_n) both exist. Let fg denote the product of the functions. Then, we have:
(fg)_{x_i}(a_1,a_2,\dots,a_n) =f_{x_i}(a_1,a_2,\dots,a_n)g(a_1,a_2,\dots,a_n) + f(a_1,a_2,\dots,a_n)g_{x_i}(a_1,a_2,\dots,a_n)
generic point, named functions Suppose f,g are both functions of variables x_1,_2,\dots,x_n. Then, for any fixed i in \{ 1,2,3,\dots,n \}: (fg)_{x_i}(x_1,x_2,\dots,x_n) =f_{x_i}(x_1,x_2,\dots,x_n)g(x_1,x_2,\dots,x_n) + f(x_1,x_2,\dots,x_n)g_{x_i}(x_1,x_2,\dots,x_n)
These hold wherever the right side expressions make sense (see concept of equality conditional to existence of one side).
generic point, named functions, point-free notation Suppose f,g are both functions of variables x,y. Then, for any fixed i in \{ 1,2,3,\dots,n \}: (fg)_{x_i} = f_{x_i}g + fg_{x_i}
These hold wherever the right side expressions make sense (see concept of equality conditional to existence of one side).

Statement for directional derivatives

Version type Statement
specific point, named functions Suppose f,g are both real-valued functions of a vector variable \overline{x}. Suppose \overline{u} is a unit vector. Suppose \overline{x_0} is a point in the domain of both functions. Then, we have the following product rule for directional derivatives:
\! \nabla_{\overline{u}}(fg)(\overline{x_0}) =  \nabla_{\overline{u}}(f)(\overline{x_0})g(\overline{x_0}) + f(\overline{x_0})\nabla_{\overline{u}}(g)(\overline{x_0})
generic point, named functions Suppose f,g are both real-valued functions of a vector variable \overline{x}. Suppose \overline{u} is a unit vector. Then, we have the following product rule for directional derivatives wherever the right side expression makes sense (see concept of equality conditional to existence of one side):
\! \nabla_{\overline{u}}(fg)(\overline{x}) = \nabla_{\overline{u}}(f)(\overline{x})g(\overline{x}) + f(\overline{x})\nabla_{\overline{u}}(g)(\overline{x}).
generic point, named functions, point-free notation Suppose f,g are both real-valued functions of a vector variable \overline{x}. Suppose \overline{u} is a unit vector. Then, we have the following product rule for directional derivatives wherever the right side expression makes sense (see concept of equality conditional to existence of one side):
\! \nabla_{\overline{u}}(fg) =  \nabla_{\overline{u}}(f)g + f\nabla_{\overline{u}}(g).


Statement for gradient vectors

Version type Statement
specific point, named functions Suppose f,g are both real-valued functions of a vector variable \overline{x}. Suppose \overline{x_0} is a point in the domain of both functions. Then, we have the following product rule for gradient vectors:
\! \nabla(fg)(\overline{x_0}) =  g(\overline{x_0}) \nabla (f)(\overline{x_0}) + f(\overline{x_0})\nabla (g)(\overline{x_0}). Note that the products on the right side are scalar-vector multiplications.
generic point, named functions Suppose f,g are both real-valued functions of a vector variable \overline{x}. Then, we have the following product rule for gradient vectors wherever the right side expression makes sense (see concept of equality conditional to existence of one side):
\! \nabla(fg)(\overline{x})=  g(\overline{x}) \nabla (f)(\overline{x}) + f(\overline{x})\nabla (g)(\overline{x}). Note that the products on the right side are scalar-vector function multiplications.
generic point, named functions, point-free notation Suppose f,g are both real-valued functions of a vector variable \overline{x}. Then, we have the following product rule for gradient vectors wherever the right side expression makes sense (see concept of equality conditional to existence of one side):
\! \nabla(fg) =  g\nabla (f) + f\nabla (g). Note that the products on the right side are scalar-vector function multiplications.

Statement for multiple functions

Statement for partial derivatives

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Statement for directional derivatives

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Statement for gradient vectors

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