To this point we’ve only looked at the two partial derivatives  and .  Recall that these derivatives represent the rate of change of f as we vary x (holding y fixed) and as we vary y (holding x fixed) respectively.  We now need to discuss how to find the rate of change of f if we allow both x and y to change simultaneously.  The problem here is that there are many ways to allow both x and y to change.  For instance one could be changing faster than the other and then there is also the issue of whether or not each is increasing or decreasing.  So, before we get into finding the rate of change we need to get a couple of preliminary ideas taken care of first.  The main idea that we need to look at is just how are we going to define the changing of x and/or y. 

Let’s start off by supposing that we wanted the rate of change of f at a particular point, say .  Let’s also suppose that both x and y are increasing and that, in this case, x is increasing twice as fast as y is increasing.  So, as y increases one unit of measure x will increase two units of measure. 

To help us see how we’re going to define this change let’s suppose that a particle is sitting at  and the particle will move in the direction given by the changing x and y.  Therefore, the particle will move off in a direction of increasing x and y and the x coordinate of the point will increase twice as fast as the y coordinate.  Now that we’re thinking of this changing x and y as a direction of movement we can get a way of defining the change.  We know from Calculus II that vectors can be used to define a direction and so the particle, at this point, can be said to be moving in the direction,

Since this vector can be used to define how a particle at a point is changing we can also use it describe how x and/or y is changing at a point.  For our example we will say that we want the rate of change of f in the direction of .  In this way we will know that x is increasing twice as fast as y is.  There is still a small problem with this however.  There are many vectors that point in the same direction.  For instance all of the following vectors point in the same direction as .

We need a way to consistently find the rate of change of a function in a given direction.  We will do this by insisting that the vector that defines the direction of change be a unit vector.  Recall that a unit vector is a vector with length, or magnitude, of 1.  This means that for the example that we started off thinking about we would want to use

since this is the unit vector that points in the direction of change.

For reference purposes recall that the magnitude or length of the vector  is given by,

For two dimensional vectors we drop the c from the formula.

Sometimes we will give the direction of changing x and y as an angle.  For instance, we may say that we want the rate of change of f in the direction of .  The unit vector that points in this direction is given by,

Okay, now that we know how to define the direction of changing x and y its time to start talking about finding the rate of change of f in this direction.  Let’s start off with the official definition.

Definition

So, the definition of the directional derivative is very similar the definition of partial derivatives.  However, in practice this can be a very difficult limit to compute so we need an easier way of taking directional derivatives.  It’s actually fairly simple to derive an equivalent formula for taking directional derivatives.

To see how we can do this let’s define a new function of a single variable,

where , , a, and b are some fixed numbers.  Note that this really is a function of a single variable now since z is the only letter that is not representing a fixed number.

Then by the definition of the derivative for functions of a single variable we have,

and the derivative at  is given by,

If we now substitute in for  we get,

So, it looks like we have the following relationship.

Now, let’s look at this from another perspective.  Let’s rewrite  as follows,

We can now use the chain rule from the previous section to compute,

So, from the chain rule we get the following relationship.

If we now take  we will get that  and  (from how we defined x and y above) and plug these into (2) we get,

Now, simply equate (1) and (3) to get that,

If we now go back to allowing x and y to be any number we get the following formula for computing directional derivatives.

This is much simpler than the limit definition.  Also note that this definition assumed that we were working with functions of two variables.  There are similar formulas that can be derived by the same type of argument for functions with more than two variables.  For instance, the directional derivative of  in the direction of the unit vector  is given by,

Let’s work a couple of examples.