12.6 The Gradient and Directional Derivatives 
Suppose that you are hiking in rugged terrain. You can think of your altitude at the point given by longitude
Suppose we want to find the instantaneous rate of change of


Notice that this limit resembles the definition of partial derivative, except that in this case, both variables may change. Further, you should observe that the directional derivative in the direction of the positive 


For convenience, we define the gradient of a function to be the vectorvalued function whose components are the firstorder partial derivatives of 

Observe that, using the gradient, we can write a directional derivative as the dot product of the gradient and the unit vector in the direction of interest, as follows. For any unit
vector ,


Theorem 6.2 makes it easy to compute directional derivatives. Further, writing directional derivatives as a dot product has many important consequences, one of which we see in the following example. 

A graphical interpretation of the directional derivatives in example 6.2 is given below. Suppose we intersect the surface
Another way of viewing the directional derivative graphically is with level curves. 

Keep in mind that the directional derivative gives the rate of change of a function in a given direction. In this case, it's reasonable to ask in what direction a given function has its maximum or minimum rate of increase. In order to answer such questions, you must first recall from Theorem 3.2 in Chapter 10 that for any two vectors
Notice now that the maximum value of occurs when 

In using Theorem 6.3, remember that the directional derivative corresponds to the rate of change of the function 

Notice that the direction of maximum increase in example 6.3 points away from the origin, since the displacement vector from


Most of the results of this section extend easily to functions of any number of variables. 

As was the case for functions of two variables, the gradient gives us a simple representation of directional derivatives in three dimensions. 

As in two dimensions, we have that


Recall that for any constant


We refer to the line through
In the following example, we illustrate the use of the gradient at a point to find the tangent plane to a surface at that point. 

Recall that in section 12.4, we found that a normal vector to the tangent plane to the surface 
Just as it is important to constantly think of ordinary derivatives as slopes of tangent lines and as instantaneous rates of change, it is crucial to keep in mind at all times the interpretations of gradients. 
