Properties of the Directional Derivatives and Gradient Vector

We know that  carries important information about the original function . In one example we saw that  tells us how steep the graph of  is; in another we saw that  tells us the velocity of an object if  tells us the position of the object at time x. As we said earlier, this same mathematical idea is useful whenever  represents some changing quantity and we want to know something about how it changes, or roughly, the “rate” at which it changes. Most functions encountered in practice are built up from a small collection of “primitive” functions in a few simple ways, for example, by adding or multiplying functions together to get new, more complicated functions. To make good use of the information provided by  we need to be able to compute it for a variety of such functions