# Product rule for partial differentiation

From Calculus

## 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*