Product rule for partial differentiation: Difference between revisions

Revision as of 23:04, 17 December 2011

Version type

Statement for functions of two variables

specific point, named function

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})

uppose 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})

Fill this in later

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@@ Line 3: / Line 3: @@
 ===Statement for partial derivatives===
-{{fillin}}
+{| class="sortable" border="1"
+! Version type !! Statement for functions of two variables
+|-
+| 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>
+|}
 ===Statement for directional derivatives===