Product rule for partial differentiation: Difference between revisions

Revision as of 04:46, 2 April 2012

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 $f g$ denote the product of the functions. Then, we have: $(f g)_{x} (x_{0}, y_{0}) = f_{x} (x_{0}, y_{0}) g (x_{0}, y_{0}) + f (x_{0}, y_{0}) g_{x} (x_{0}, y_{0})$ Suppose the partial derivatives $f_{y} (x_{0}, y_{0})$ and $g_{y} (x_{0}, y_{0})$ both exist. Then, we have: $(f g)_{y} (x_{0}, y_{0}) = f_{y} (x_{0}, y_{0}) g (x_{0}, y_{0}) + f (x_{0}, y_{0}) g_{y} (x_{0}, y_{0})$
generic point, named functions	Suppose $f, g$ are both functions of variables $x, y$ . $(f g)_{x} (x, y) = f_{x} (x, y) g (x, y) + f (x, y) g_{x} (x, y)$ $(f g)_{y} (x, y) = f_{y} (x, y) g (x, y) + f (x, y) g_{y} (x, y)$ 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 $x, y$ . $(f g)_{x} = f_{x} g + f g_{x}$ $(f g)_{y} = f_{y} g + f g_{y}$ 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 $\bar{x}$ . Suppose $\bar{u}$ is a unit vector. Suppose $\bar{x_{0}}$ is a point in the domain of both functions. Then, we have the following product rule for directional derivatives: $\nabla_{\bar{u}} (f g) (\bar{x_{0}}) = g (\bar{x_{0}}) \nabla_{\bar{u}} (f) (\bar{x_{0}}) + f (\bar{x_{0}}) \nabla_{\bar{u}} (g) (\bar{x_{0}})$
generic point, named functions	Suppose $f, g$ are both real-valued functions of a vector variable $\bar{x}$ . Suppose $\bar{u}$ 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): $\nabla_{\bar{u}} (f g) (\bar{x}) = g (\bar{x}) \nabla_{\bar{u}} (f) (\bar{x}) + f (\bar{x}) \nabla_{\bar{u}} (g) (\bar{x})$ .
generic point, named functions, point-free notation	Suppose $f, g$ are both real-valued functions of a vector variable $\bar{x}$ . Suppose $\bar{u}$ 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): $\nabla_{\bar{u}} (f g) = g \nabla_{\bar{u}} (f) + f \nabla_{\bar{u}} (g)$ .

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

Statement for gradient vectors

Version type	Statement
specific point, named functions	Suppose $f, g$ are both real-valued functions of a vector variable $\bar{x}$ . Suppose $\bar{x_{0}}$ is a point in the domain of both functions. Then, we have the following product rule for gradient vectors: $\nabla (f g) (\bar{x_{0}}) = g (\bar{x_{0}}) \nabla (f) (\bar{x_{0}}) + f (\bar{x_{0}}) \nabla (g) (\bar{x_{0}})$ . 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 $\bar{x}$ . 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): $\nabla_{\bar{u}} (f g) = g (\bar{x}) \nabla (f) (\bar{x}) + f (\bar{x}) \nabla (g) (\bar{x})$ . Note that the products on the right side are scalar-vector multiplications.
generic point, named functions, point-free notation	Suppose $f, g$ are both real-valued functions of a vector variable $\bar{x}$ . 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): $\nabla (f g) = g \nabla (f) + f \nabla (g)$ . Note that the products on the right side are scalar-vector 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

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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>
 |-
-| 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]]).
 |}
@@ Line 20: / Line 20: @@
 | 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}) =  g(\overline{x_0})\nabla_{\overline{u}}(f)(\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}) = g(\overline{x})\nabla_{\overline{u}}(f)(\overline{x}) + f(\overline{x})\nabla_{\overline{u}}(g)(\overline{x})</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 (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x}) = g(\overline{x})\nabla_{\overline{u}}(f)(\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) =  g\nabla_{\overline{u}}(f) + f\nabla_{\overline{u}}(g)</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 (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla_{\overline{u}}(fg) =  g\nabla_{\overline{u}}(f) + f\nabla_{\overline{u}}(g)</math>.
 |}
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 | 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>\! \nabla_{\overline{u}}(fg)=  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 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 (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla_{\overline{u}}(fg)=  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 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) =  g\nabla (f) + f\nabla (g)</math>. Note that the products on the right side are scalar-vector 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 (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 multiplications.
 |}