First-order partial derivatives of functions with two variables. Have a look!!

Dear friends, today’s topic is first-order partial derivatives of functions with two variables. In general, we all have studied partial differentiation during high school. So this is more like a re-visit to the good old topic.

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

If you are looking for more in partial differentiation, do check in:

**First-order partial derivatives of functions with three variables**

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

**Partial derivatives of inverse functions**

**Change of variables in first-order partial differentiation**

**How to use partial differentiation to find out the nature of stationary points**

First-order partial derivative of two variables

Suppose is a function of two variables and .

Let me choose .

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

The mathematical symbol of the first-order partial derivative of with respect to is .

Also, the mathematical symbol of the first-order partial derivative of with respect to is .

Next, I’ll determine the value of .

This means I’ll differentiate partially with respect to . At that time, the other variable will act as a constant.

Also, for partial differentiation, the rules are the same as ordinary differentiation.

See more: **Rules for ordinary differentiation**

So will be

This gives

In the same way will be

Therefore I can say that

And there are many more examples like that.

So now I’ll give you some examples.

#### Examples of the first-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 first-order partial derivatives. Soon I’ll write another post on second-order partial derivatives.

So, here it is.

Here the given function is

I have to get first-order partial derivatives of .

That means I need to find out the values of and .

I’ll start with .

First of all, I’ll differentiate partially with respect to .

This gives

Now for , the variable will act as a constant.

So it gives

Now I’ll get the value of .

Here again I’ll differentiate partially with respect to .

This gives

Now for , the variable will act as a constant.

So it gives

Hence I can conclude that the first 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 first-order partial derivatives. Soon I’ll write another post on second-order partial derivatives.

So, here it is.

Here the given function is

I have to get first-order partial derivatives of .

That means I need to find out the values of and .

I’ll start with .

First of all, I’ll differentiate partially with respect to .

This gives

Now for , the variable will act as a constant.

So it gives

Now I’ll get the value of .

Here again, I’ll differentiate partially with respect to .

This gives

Now for , the variable will act as a constant.

So it gives

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

Hence I can conclude that this is the answer to the given example.

Now I am going to give two more examples. Also, I don’t have a clue about the references to these two examples.

##### Example 3

Find all of the first order partial derivatives for the following functions:

(c)

##### Solution

Here the given function is As I can see, is a function of two variables and . And I have to get first-order partial derivatives of .

So that means I need to find out the values of and .

But before that, I’ll simplify the function .

###### Step 1

For example, I can write as . And this is because of .

Also, I can rewrite as .

Again, I can rewrite as since .

So this means the function becomes

Now I’ll differentiate partially with respect to and .

###### Step 2

If I differentiate partially with respect to , it gives

Now for , the variable will act as a constant.

So it gives

Next, I’ll simplify it to get

Now I’ll get the value of .

So I’ll differentiate partially with respect to .

This gives

Now for , the variable will act as a constant.

So it gives

Then I’ll simplify it to get

Hence I can conclude that the first partial order derivatives of the function are and

Therefore I can say that this is the answer to the given example.

Now I am going to give another example.

##### Example 4

Find all of the first order partial derivatives for the following functions:

(d) .

##### Solution

Here the given function is

I have to get first-order partial derivatives of .

That means I need to find out the values of and .

I’ll start with .

First of all, I’ll differentiate partially with respect to .

###### Step 1

And this gives

Now for , the variable will act as a constant.

Also, I can see the function is a product of two functions and .

So in this case, I’ll use the **product rule for differentiation** to differentiate the function .

Therefore it will be

Next, I’ll simplify it to get

Now I’ll do further partial differentiation to get

Then I’ll simplify it to get

Now I’ll get the value of .

###### Step 2

Here again, I’ll differentiate partially with respect to .

This gives

Now for , the variable will act as a constant.

Then it gives

Next, I’ll simplify it to get

So this becomes

Then I can rearrange them to get

Hence I can conclude that the first partial order derivatives of the function are and .

Therefore I can conclude that this is the answer to the given example.

Dear friends, this is the end of 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!!

Slumzz Napo says

Thanks much. Its very helpful.