Product rule for partial differentiation: Difference between revisions

Revision as of 23:39, 17 December 2011

Statement for two functions

Statement for partial derivatives

Version type	Statement for functions of two variables
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 $x,y$ . $(f\cdot g)_{x}(x,y)=f_{x}(x,y)g(x,y)+f(x,y)g_{x}(x,y)$ $(fg)_{y}(x,y)=f_{y}(x,y)g(x,y)+f(x,y)g_{y}(x,y)$ These hold wherever the right side expressions make sense.
generic point, named functions, point-free notation	Suppose $f,g$ are both functions of variables $x,y$ . $(fg)_{x}=f_{x}g+fg_{x}$ $(fg)_{y}=f_{y}g+fg_{y}$ These hold wherever the right side expressions make sense.

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: $\!\nabla _{\overline {u}}(fg)({\overline {x_{0}}})=f({\overline {x_{0}}})\nabla _{\overline {u}}(g)({\overline {x_{0}}})+g({\overline {x_{0}}})\nabla _{\overline {u}}(f)({\overline {x_{0}}})$
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: $\!\nabla _{\overline {u}}(fg)({\overline {x}})=f({\overline {x}})\nabla _{\overline {u}}(g)({\overline {x}})+g({\overline {x}})\nabla _{\overline {u}}(f)({\overline {x}})$ .
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. Suppose ${\overline {x_{0}}}$ is a point in the domain of both functions. Then, we have the following product rule for directional derivatives: $\!\nabla _{\overline {u}}(fg)=f\nabla _{\overline {u}}(g)+g\nabla _{\overline {u}}(f)$ .

The rule applies at all points where the right side make sense.

Statement for gradient vectors

Fill this in later

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

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 ! Version type !! Statement for functions of two variables
 |-
-| 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. Then, we have:<br><math>(f \cdot g)_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>(f \cdot g)_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>(f \cdot g)_x(x,y) =f_x(x,y)g(x,y) + f(x,y)g_x(x,y)</math><br><math>(f \cdot g)_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>(f \cdot g)_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, point-free notation || Suppose <math>f,g</math> are both functions of variables <math>x,y</math>. <br><math>(f \cdot g)_x =f_x \cdot g + f \cdot g_x</math><br><math>(f \cdot g)_y = f_y\cdot g + f\cdot g_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.
 |}
 ===Statement for directional derivatives===
-Suppose <math>f,g</math> are both real-valued functions of many variables. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s:
+{| class="sortable" border="1"
+! Version type !! Statement
-<math>\nabla_{\overline{u}}(f \cdot g)|_{\overline{x}} =  f \nabla_{\overline{u}}(g) + g \nabla_{\overline{u}}(f)</math>
+|-
+| 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}) =  f(\overline{x_0})\nabla_{\overline{u}}(g)(\overline{x_0}) + g(\overline{x_0}) \nabla_{\overline{u}}(f)(\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:<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x}) =  f(\overline{x})\nabla_{\overline{u}}(g)(\overline{x}) + g(\overline{x}) \nabla_{\overline{u}}(f)(\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. 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) =  f\nabla_{\overline{u}}(g) + g\nabla_{\overline{u}}(f)</math>.
+|}
 The rule applies at all points where the right side make sense.