see: multi-variable derivative, gradient, and directional derivative

Multi-Dimensional Derivative and Gradient

Gradient only works for functions that returns one output (). In that case, the multi-variable derivative matrix will be a row vector:

This is exactly the transpose of the gradient:

We can view the multi-variable derivative as a linear transformation acting on a vector, while the gradient is simply a regular vector.

Directional Derivative

The “directional derivative” can be defined via the as dot product from gradient to the direction, and we can also derive it directly from the multi-variable derivative matrix:

where is a unit vector.

Different Ways of Thinking

The use of different terminology and different math construct for the same concept demonstrate two different ways of thinking:

vector analysislinear algebra
directional derivativerate of change of output
direction vector any vector of rates of change of input
Require to be unit vector & same units on all axisindifferent to lengths and units

Reference