# Directional derivative

This article describes an analogue for functions of multiple variables of the following term/fact/notion for functions of one variable: derivative

## Contents

## Definition at a point

### Generic definition

Suppose is a function of many variables. Consider the domain of as a subset of Euclidean space. Fix a direction in this space and a point in the domain. Then, the **directional derivative** at the point in the direction is the derivative of the function with respect to movement of the point along that direction, at the specific point.

### For a function of two variables

Suppose is a function of two variables . Suppose is a unit vector (i.e., we have ). Suppose is a point in the domain of We define the **directional derivative** of at in the direction of as follows.

Item | Value |
---|---|

Notation | or |

Definition as a limit | |

Definition as an ordinary derivative |

### For a function of multiple variables

Suppose is a function of variables . Suppose is a unit vector (i.e., we have ). Suppose is a point in the domain of . The **directional derivative** of at in the direction of is defined as follows.

Item | Value |
---|---|

Notation | or |

Definition as a limit | |

Definition as an ordinary derivative |

### For a function of multiple variables in vector notation

Suppose is a function of a vector variable . Suppose is a unit vector and is a point in the domain of . The directional derivative of at in the direction of is denoted and defined as below.

Item | Value |
---|---|

Notation | or |

Definition as a limit | |

Definition as an ordinary derivative |

## Definition as a function

### Generic definition

Suppose is a function of many variables. Consider the domain of as a subset of Euclidean space. Fix a direction in this space. Then, the **directional derivative** in the direction is the function sending a point in the domain of to the derivative of the function with respect to movement of the point along that direction.

### For a function of two variables

Suppose is a function of two variables , with domain a subset of . Suppose is a unit vector (i.e., we have ). Then, the **directional derivative** in the direction of is a function with domain a subset of the domain of , defined as the function that sends any point in the domain of to the directional derivative of in the direction of at the point.

Item | Value |
---|---|

Notation | or |

Definition as a limit | |

Definition as a partial derivative | . Note that we need to use a partial derivative because are now variable as we are not doing this at a single point. |

### For a function of multiple variables

Suppose is a function of variables . Suppose is a unit vector (i.e., we have ). We define and denote the directional derivative as below.

Item | Value |
---|---|

Notation | or |

Definition as a limit | |

Definition as an ordinary derivative | . Note that we need to use a partial derivative because are now variable as we are not doing this at a single point. |

### For a function of multiple variables in vector notation

Suppose is a function of a vector variable . Suppose is a unit vector. We define and denote the directional derivative of in the direction of below.

Item | Value |
---|---|

Notation | or |

Definition as a limit | |

Definition as an ordinary derivative | . Note that we need to take the partial derivative instead of the ordinary derivative because the coordinates of themselves are not fixed, as we are doing this at a generic rather than a fixed point. |

## Graphical interpretation

### For a function of two variables

`For further information, refer: graph of a function of two variables`

The directional derivative in the direction of a unit vector at a point can be determined as follows: first, intersect the graph of the function with the plane . This plane is perpendicular to the -plane and its intersection with the -plane is the line through in the direction of the unit vector .

This intersection can be thought of as the graph of a function of one variable, where the point is treated as the origin, the direction is the independent variable axis, and the -axis direction is the dependent variable axis. Now, the directional derivative is the slope of this graph for dependent variable value of 0.

## Relation with gradient vector

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

at a point, in vector notation (multiple variables) | Suppose is a function of a vector variable . Suppose is a unit vector and is a point in the domain of . Suppose that the gradient vector of at exists. We denote this gradient vector by . Then, we have the following relationship: The right side here is the dot product of vectors. |

generic point, in vector notation (multiple variables) | Suppose is a function of a vector variable . Suppose is a unit vector. We then have: The right side here is a dot product of vectors. The equality holds whenever the right side makes sense. |

generic point, point-free notation (multiple variables) | Suppose is a function of a vector variable . Suppose is a unit vector. We then have: The right side here is a dot product of vector-valued functions (the constant function and the gradient vector of ). The equality holds whenever the right side makes sense. |

## Relation with partial derivatives

### Partial derivatives as directional derivatives

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

Generic | For a function of many variables , the partial derivative can be thought of as the directional derivative in the direction of the unit vector where and all other coordinates are zero. |

### Directional derivative in terms of partial derivatives

If the gradient vector at a point exists, then it is a vector whose coordinates are the corresponding partial derivatives of the function. Thus, *conditional to the existence of the gradient vector*, we have that:

In particular, for a function of two variables and a unit vector , we have the following, assuming that the gradient vector exists (i.e., that the function is differentiable):