Product rule for partial differentiation: Difference between revisions

Revision as of 23:09, 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. Then, we have: $(f \cdot g)_{x} (x_{0}, y_{0}) = f_{x} (x_{0}, y_{0}) g (x_{0}, y_{0}) + f (x_{0}, y_{0}) g_{x} (x_{0}, y_{0})$ Suppose the partial derivatives $f_{y} (x_{0}, y_{0})$ and $g_{y} (x_{0}, y_{0})$ both exist. Then, we have: $(f \cdot g)_{y} (x_{0}, y_{0}) = f_{y} (x_{0}, y_{0}) g (x_{0}, y_{0}) + f (x_{0}, y_{0}) g_{y} (x_{0}, y_{0})$
generic point, named functions	Suppose $f, g$ are both functions of variables $x, y$ . $(f \cdot g)_{x} (x, y) = f_{x} (x, y) g (x, y) + f (x, y) g_{x} (x, y)$ $(f \cdot g)_{y} (x, y) = f_{y} (x, y) g (x, y) + f (x, y) g_{y} (x, y)$ These hold wherever the right side expressions make sense.
generic point, named functions, point-free notation	Suppose $f, g$ are both functions of variables $x, y$ . $(f \cdot g)_{x} = f_{x} \cdot g + f \cdot g_{x}$ $(f \cdot g)_{y} = f_{y} \cdot g + f \cdot g_{y}$ These hold wherever the right side expressions make sense.

Statement for directional derivatives

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Statement for gradient vectors

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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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 ! Version type !! Statement for functions of two variables
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-| specific point, named function || 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>uppose 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. 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>
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+| 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.
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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>. <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.
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