First-order partial derivatives of functions with two variables. Dear friends, today’s topic is first-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!!

### First-order partial derivatives of functions with 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 the 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.

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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