Difference between revisions of "Product rule for partial differentiation"
From Calculus
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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. | | 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. | ||
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− | | 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>\! \ | + | | 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. |
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| 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. | | 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. |
Revision as of 16:02, 2 April 2012
Contents
Statement for two functions
Statement for partial derivatives
Version type | Statement for functions of two variables |
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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 directional derivatives
Version type | Statement |
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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): . |
The rule applies at all points where the right side make sense.
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
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Statement for directional derivatives
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Statement for gradient vectors
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