# Relation between gradient vector and directional derivatives

From Calculus

## Statement of relation

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. |