# Difference between revisions of "Product rule for partial differentiation"

From Calculus

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! Version type !! Statement | ! Version type !! Statement | ||

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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{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}) = | + | | 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}) = \nabla_{\overline{u}}(f)(\overline{x_0})g(\overline{x_0}) + f(\overline{x_0})\nabla_{\overline{u}}(g)(\overline{x_0})</math> |

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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>. 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}) = | + | | 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}) = + \nabla_{\overline{u}}(f)(\overline{x})g(\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) = | + | | 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) = \nabla_{\overline{u}}(f)g + f\nabla_{\overline{u}}(g)</math>. |

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## Revision as of 04:33, 2 April 2012

## Contents

## Statement for two functions

### Statement for partial derivatives

Version type | Statement for functions of two variables |
---|---|

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. |

generic point, named functions, point-free notation | Suppose are both functions of variables . These hold wherever the right side expressions make sense. |

### Statement for directional derivatives

Version type | Statement |
---|---|

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: . |

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: . |

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: . Note that the products on the right side are scalar-vector 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: . 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*