First-order partial derivative of functions with three variables. Dear friends, today’s topic is the first-order partial derivative of functions with three variables.
First-order partial derivatives of functions with three variables_compressed
The first-order partial derivative of functions with three variables
In my earlier post on the first-order partial derivative of functions with two variables, I have discussed in great detail how things work.
So here without much ado, I’ll start with some examples.
Examples of the First-order partial derivative of functions with three 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)* “If express in its simplest form
”
Solution
Here the given function is
And, I have to find out the value of
So I’ll start with .
Step 1
First of all, I’ll differentiate partially with respect to
to get
(1)
Next, I’ll differentiate partially with respect to
to get
(2)
Finally, I’ll differentiate partially with respect to
to get
(3)
So, now I’ll find out the value of
Step 2
For that, I’ll use the values of and
from equations (1), (2) and (3) respectively.
Thus it will be
Therefore I get
Hence I can conclude that this is the answer to this example.
Now I’ll go to the next example.
Example 2
According to Stroud and Booth (2013)* “If , show that
”
Solution
In this example, the given function is
And, I have to prove that
So I’ll start with .
Step 1
First of all, I’ll differentiate partially with respect to
to get
Therefore I get
Thus I can say will be
(4)
In the same way I can also get the value of .
Thus will be
Therefore I can say will be
(5)
Similarly I can also get the value of .
Thus will be
Therefore I can say will be
(6)
So, now I’ll find out the value of
Step 2
For that, I’ll use the values of and
from equations (4), (5) and (6) respectively.
Thus it will be
Hence I can conclude that I have proved .
This is the answer to this 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!!
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