Hello friends, today I will talk about directional derivative.

### Directional derivative

Here I don’t go much into the theory. Instead, I will try to explain it in a simpler way.

##### Few things to know about directional derivative

- Directional derivative only happens to some scalar field or say some function.

- Directional derivative has the notation with as a scalar field or function.

- is the projection of the gradient vector of on the given unit vector in the direction of .

#### Example of directional derivative

Note: This example does not belong to me. I have chosen this from a book. I have also given the due reference at the end of the post.

So here it is.

##### Example

According to Stroud and Booth (2011)* “Find the directional derivative of at the point in the direction of .”

##### Solution

In this example, the given surface is

To get the directional derivative of this surface, I will follow certain steps. The first step is to get the gradient of this surface.

In one of my earlier posts on vector analysis, I have explained how to get the gradient of any surface.

Here also I will do the same.

To get the value of grad , I will start with the partial differentiations.

###### Step 1

So will be

Now I’ll get Thus it will be

Similarly, will be

Thus the gradient of is

###### Step 2

Now I will find out the value of the gradient vector at the point

Thus at the point the value of the gradient vector will be

Next I have to find out the unit vector in the direction of the given vector

###### Step 3

In order to find out the unit vector , first of all I will determine the magnitude of the vector

Thus the magnitude of the vector is

Therefore the unit vector in the direction of the given vector is

###### Step 4

As a next step, now I will get the directional derivative.

According to the formula, directional derivative of the surface is .

Here both and are vectors. Thus is the scalar product of two vectors.

Therefore in this case it will be

Hence I can conclude that the directional derivative of at the point in the direction of is

This is the answer to this example.

So here ends my example.

Dear friends, this is the end of my today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

###### *Reference: K. A. Stroud and Dexter J. Booth (2011): Advanced engineering mathematics, Industrial Press, Inc.; 5th Edition (March 8, 2011), Chapter: Vector analysis 1, Further problems 22, p. 815, Q. No. 11 (Example 1).

d φ ⁄ ds = **a**. grad φ