Second-order partial derivatives of functions with two variables. Dear friends, today’s topic is second-order partial derivatives of functions with two variables. In general, we all have studied partial differentiation during our high school.

So this is more like a re-visit to the good old topic. Have a look!!

### Second-order partial derivatives of functions with two variables

Suppose is a function of two real variables and .

Let me choose .

Now I want to get the second order partial derivative of with respect to both and .

For that, first of all, I have to get the first-order partial derivatives of with respect to both and .

In my earlier post on first-order partial derivatives of functions with two variables, I have discussed in detail how to work on that.

So in this example, the first-order partial derivative of with respect to will be

Similarly, the first-order partial derivative of with respect to will be

Now, for the second order partial derivatives of with respect to , there will be two derivatives.

One is . Another one is .

Thus in this example, will be

Similarly, will be

In a similar way, there will also be two second order partial derivatives of with respect to .

These are and respectively.

So in this example, will be

In the same way, will be

Now I’ll give you some examples.

#### Examples of the second-order partial derivatives of functions with two variables

Disclaimer: None of these examples is mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here are the examples.

##### Example 1

According to Stroud and Booth (2013)* “Find all first and second partial derivatives of the following: ”

##### Solution

In this post, I’ll only do the second-order partial derivatives. I have already worked out the first-order partial derivatives of the function in my earlier post.

So, here it is.

Here the given function is

Now the first-order partial derivatives of the function with respect to and are

(1)

and

(2)

respectively.

Next, I’ll start with the second order partial derivatives of the function with respect to .

###### Step 1

For the second order partial derivatives of with respect to , there will be two derivatives.

One is . Another one is .

First I’ll get the value of .

For that, I’ll differentiate equation (1) partially with respect to .

This gives

Now I’ll get the value of .

For that, I’ll differentiate equation (1) partially with respect to .

So this will be

Next I’ll get the second order partial derivatives of the function with respect to .

###### Step 2

For the second order partial derivatives of with respect to , there will be two derivatives.

One is . Another one is .

First I’ll get the value of .

For that, I’ll differentiate equation (2) partially with respect to .

This gives

Now I’ll get the value of .

For that, I’ll differentiate equation (2) partially with respect to .

So this will be

Hence I can conclude that the second partial derivatives of the function are , , and

This is the answer to this example.

Now I’ll go to the next example.

##### Example 2

According to Stroud and Booth (2013)* “Find all first and second partial derivatives of the following: ”

##### Solution

Like my first example, here also I’ll only do the second-order partial derivatives. I have already worked out the first-order partial derivatives of the function in my earlier post.

Here the given function is

Now the first-order partial derivatives of the function with respect to and are

(3)

and

(4)

respectively.

Next, I’ll start with the second order partial derivatives of the function with respect to .

###### Step 1

For the second order partial derivatives of with respect to , there will be two derivatives.

One is . Another one is .

First I’ll get the value of .

For that, I’ll differentiate equation (3) partially with respect to .

This gives

Now I’ll get the value of .

For that, I’ll differentiate equation (3) partially with respect to .

So this will be

Next I’ll get the second order partial derivatives of the function with respect to .

###### Step 2

For the second order partial derivatives of with respect to , there will be two derivatives.

One is . Another one is .

First I’ll get the value of .

For that, I’ll differentiate equation (4) partially with respect to .

This gives

Now I’ll get the value of .

For that, I’ll differentiate equation (4) partially with respect to .

So this will be

Therefore my conclusion is that the second partial derivatives of the function are

Hence this is the answer to the given 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!!

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