Directional Derivatives

Recall the geometric interpretation of the partial derivative that we can find the slope of the tangent along the fixed or directions. We now wish to generalize this.

What about the slope in the direction of a different unit vector ?
This is called the 'directional derivative':

Since is a function dependent only on (we "know" the rest of the variables), we can make it a single variable function:

where , and .

Now, using the chain rule for paths:

(Dot product is also a projection of sorts)