Difference between revisions of "Product rule for partial differentiation"
From Calculus
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==Statement for two functions== | ==Statement for two functions== | ||
− | ===Statement for partial derivatives=== | + | ===Statement for partial derivatives for functions of two variables=== |
+ | |||
+ | The derivatives used here are [[partial derivative]]s. | ||
{| class="sortable" border="1" | {| class="sortable" border="1" | ||
− | ! Version type !! Statement | + | ! Version type !! Statement |
|- | |- | ||
| 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> | ||
|- | |- | ||
− | | generic point, named functions || Suppose <math>f,g</math> are both functions of variables <math>x,y</math>. <br><math>(fg)_x(x,y) =f_x(x,y)g(x,y) + f(x,y)g_x(x,y)</math><br><math>(fg)_y(x,y) = f_y(x,y)g(x,y) + f(x,y)g_y(x,y)</math><br>These hold wherever the right side expressions make sense. | + | | generic point, named functions || Suppose <math>f,g</math> are both functions of variables <math>x,y</math>. <br><math>(fg)_x(x,y) =f_x(x,y)g(x,y) + f(x,y)g_x(x,y)</math><br><math>(fg)_y(x,y) = f_y(x,y)g(x,y) + f(x,y)g_y(x,y)</math><br>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 <math>f,g</math> are both functions of variables <math>x,y</math>. <br><math>(f g)_x =f_xg + fg_x</math><br><math>(f g)_y = f_yg + fg_y</math><br>These hold wherever the right side expressions make sense. | + | | generic point, named functions, point-free notation || Suppose <math>f,g</math> are both functions of variables <math>x,y</math>. <br><math>(f g)_x =f_xg + fg_x</math><br><math>(f g)_y = f_yg + fg_y</math><br>These hold wherever the right side expressions make sense (see [[concept of equality conditional to existence of one side]]). |
|} | |} | ||
+ | <center>{{#widget:YouTube|id=lTFCy8V5qDc}}</center> | ||
+ | ===Statement for partial derivatives for functions of multiple variables=== | ||
+ | |||
+ | {| class="sortable" border="1" | ||
+ | ! Version type !! Statement | ||
+ | |- | ||
+ | | 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> | ||
+ | |- | ||
+ | | 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]]). | ||
+ | |- | ||
+ | | 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]]). | ||
+ | |} | ||
===Statement for directional derivatives=== | ===Statement for directional derivatives=== | ||
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! Version type !! Statement | ! Version type !! Statement | ||
|- | |- | ||
− | | specific point, named functions || Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[directional derivative]]s:<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x_0}) = | + | | specific point, named functions || Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[directional derivative]]s:<br><math>\! \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})</math> |
|- | |- | ||
− | | generic point, named functions || Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense:<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x}) = | + | | generic point, named functions || Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x}) = \nabla_{\overline{u}}(f)(\overline{x})g(\overline{x}) + f(\overline{x})\nabla_{\overline{u}}(g)(\overline{x})</math>. |
|- | |- | ||
− | | generic point, named functions, point-free notation || Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense:<br><math>\! \nabla_{\overline{u}}(fg) = | + | | generic point, named functions, point-free notation || Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla_{\overline{u}}(fg) = \nabla_{\overline{u}}(f)g + f\nabla_{\overline{u}}(g)</math>. |
|} | |} | ||
− | + | ||
===Statement for gradient vectors=== | ===Statement for gradient vectors=== | ||
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! Version type !! Statement | ! Version type !! Statement | ||
|- | |- | ||
− | | specific point, named functions || Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[gradient vector]]s:<br><math>\! \nabla(fg)(\overline{x_0}) = | + | | specific point, named functions || Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[gradient vector]]s:<br><math>\! \nabla(fg)(\overline{x_0}) = g(\overline{x_0}) \nabla (f)(\overline{x_0}) + f(\overline{x_0})\nabla (g)(\overline{x_0})</math>. Note that the products on the right side are scalar-vector multiplications. |
|- | |- | ||
− | | generic point, named functions || Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense:<br><math>\! \ | + | | generic point, named functions || Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla(fg)(\overline{x})= g(\overline{x}) \nabla (f)(\overline{x}) + f(\overline{x})\nabla (g)(\overline{x})</math>. Note that the products on the right side are scalar-vector ''function'' multiplications. |
|- | |- | ||
− | | generic point, named functions, point-free notation || Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense:<br><math>\! \nabla(fg) = | + | | generic point, named functions, point-free notation || Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla(fg) = g\nabla (f) + f\nabla (g)</math>. Note that the products on the right side are scalar-vector ''function'' multiplications. |
|} | |} | ||
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 are both functions of variables . Suppose is a point in the domain of both and . Suppose the partial derivatives and both exist. Let denote the product of the functions. Then, we have: Suppose the partial derivatives and both exist. Then, we have: |
generic point, named functions | Suppose are both functions of variables . 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 are both functions of variables . 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 are both functions of variables . Suppose is a point in the domain of both and . Fix a number in . Suppose the partial derivatives and both exist. Let denote the product of the functions. Then, we have: |
generic point, named functions | Suppose are both functions of variables . Then, for any fixed in : 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 are both functions of variables . Then, for any fixed in : 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 are both real-valued functions of a vector variable . Suppose is a unit vector. Suppose is a point in the domain of both functions. Then, we have the following product rule for directional derivatives: |
generic point, named functions | Suppose are both real-valued functions of a vector variable . Suppose 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): . |
generic point, named functions, point-free notation | Suppose are both real-valued functions of a vector variable . Suppose 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): . |
Statement for gradient vectors
Version type | Statement |
---|---|
specific point, named functions | Suppose are both real-valued functions of a vector variable . Suppose is a point in the domain of both functions. Then, we have the following product rule for gradient vectors: . Note that the products on the right side are scalar-vector multiplications. |
generic point, named functions | Suppose are both real-valued functions of a vector variable . 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): . Note that the products on the right side are scalar-vector function multiplications. |
generic point, named functions, point-free notation | Suppose are both real-valued functions of a vector variable . 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): . Note that the products on the right side are scalar-vector function multiplications. |
Statement for multiple functions
Statement for partial derivatives
Fill this in later
Statement for directional derivatives
Fill this in later
Statement for gradient vectors
Fill this in later