Suppose is a given function. We have seen that the first partial derivatives are

(1)

(2)

if these limits exist. Now if we use an arbitrary direction, say a unit vector , then both and will have changes. We define the directional derivative of with respect to an arbitrary unit vector in -plane by

(3)

if this limit exist! In fact, it can be shown that

(4)

Let us see why. Define

(5)

and by Chain Rule,

(6)

and of course

(7)

The vector used above is called the gradient of and is shown by

We want to find the directional derivative of with respect to the standard position angle at . The unit vector used is . The gradient of is

(8)

Now we compute the directional derivative of with respect to the given direction

(9)

At the point we get

(10)

From the construction of directional derivative of in direction of we get

(11)

Since maximum value of is equal to we see immediately that the maximum value of directional derivative is . On top of this, when and so must be parallel to to get the maximum directional derivative.

We want to find the direction of maximum rate of change and its value for the function at the point . We start by writing

(12)

At the direction which gives the maximum rate of change is and the maximum rate of change is

Suppose a function of two variables is given where is differentiable with respect to and and also each and are differentiable with respect to . An arbitrary level curve of this function associated to is identified by which is a curve in -plane. We can differentiate both side of this equation to get

(13)

This means

(14)

We know is a tangent vector at any given point. This dot product being equal to zero implies that a tangent vector to the level curve and the gradient vector are perpendicular to each other.

Suppose a function of three variables is given where is differentiable with respect to and each of is differentiable with respect to Then an arbitrary level surface of associated to a constant is

(15)

Similar to previous case, observation 2, we can write

(16)

Of course like before this means that the gradient vector and the tangent vector at any point are perpendicular to each other; we also see the equation of the tangent plane can be written using the gradient vector as the normal that is for the point

(17)

More, we can write equation of the normal line to the level surface at any point

(18)

To find the equation of the tangent plane to the paraboloid we define The level surface of associated to is precisely the given paraboloid. And so

(19)

is the desired tangent plane. Next we find the equation of the normal line:

(20)

To see more sample problems, visit directional derivative sample problems.