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

From Calculus

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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:<br><math>\! \nabla_{\overline{u}}(fg)= f(\overline{x})\nabla (g)(\overline{x}) + g(\overline{x}) \nabla (f)(\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:<br><math>\! \nabla_{\overline{u}}(fg)= f(\overline{x})\nabla (g)(\overline{x}) + g(\overline{x}) \nabla (f)(\overline{x})</math>. Note that the products on the right side are scalar-vector 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> | + | | 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) = f\nabla (g) + g\nabla (f)</math>. Note that the products on the right side are scalar-vector multiplications. |

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## Revision as of 04:19, 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*