Determine the differential dz. Hello friends, today I’ll show how to determine the differential dz. Have a look!!

### Determine the differential dz

Suppose, is a function of two real variables and . Now I want to determine the differential .

So the differential will be

Now here and .

Similarly, suppose is a function of three real variables and . And I want to determine the differential .

Thus the differential will be

Now here

Next, I’ll give some examples of how to determine .

#### Some examples on how to determine the differential dz

Note: None of these examples are 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 is the first example.

##### Example 1

According to Stroud and Booth (2011)* “Determine the differential of the following: .”

##### Solution

Now in this example, the function is

So in order to find out the value of differential , I’ll start with the partial differentiation of .

As I can see, the function is dependent on two variables and . So I’ll use the method for first-order partial differentiation of function with two variables.

Therefore I’ll differentiate partially with respect to to get the value of as

So this means

Next, I’ll differentiate partially with respect to to get the value of as

Thus it gives

As per the formula for the differential where and .

Hence in this example, the value of the differential will be

So this means

is the differential of the function .

Hence I can conclude that this is the solution to this example.

Now I’ll give some more examples.

So here comes my second example.

##### Example 2

According to Stroud and Booth (2011)* “Determine the differential of the following: .”

##### Solution

Now in this example, the function is

So in order to find out the value of differential , I’ll start with the partial differentiation of .

As I can see, the function is dependent on two variables and . So I’ll use the same method as in example 1.

Therefore I’ll differentiate partially with respect to to get the value of as

So this means

Next, I’ll differentiate partially with respect to to get the value of as

Thus it gives

Hence in this example, the value of the differential will be

Hence I can conclude that this is the solution to this example.

Now I’ll give one more example.

So here it goes.

##### Example 3

According to Stroud and Booth (2011)* “Determine the differential dz of the following: .”

##### Solution

Now in this example, the function is

So in order to determine the differential dz, I’ll start with the partial differentiation of .

As I can see, the function is dependent on three variables and . So I’ll use the method for first-order partial differentiation of function with three variables.

Therefore I’ll differentiate partially with respect to to get the value of as

So this means

Next, I’ll differentiate partially with respect to to get the value of as

Thus it gives

Then I’ll differentiate partially with respect to to get the value of as

So this means

As per the formula for the differential where and .

Hence in this example, the value of the differential will be

So this means

is the differential of the function .

Hence I can conclude that this is the solution to this example.

Dear friends, this is the end of my today’s post on how to determine the differential dz. 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!!

## Leave a Reply