Partial derivatives of inverse functions. Hello friends, today it’s about partial derivatives of inverse functions.

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### Partial derivatives of inverse functions

Well, suppose I have two functions, say and . And both of these two functions are dependent on the same variables, say, and .

So I can say that and .

Now it’s not a problem to get the partial derivative of or with respect to or .

But if I have to get the partial derivative of with respect to or , then I need to use the formulas for the partial derivatives of inverse functions.

In the same way, it’s also true for the partial derivatives of with respect to and .

So here I’ll summarise these formulas for and .

###### Summary of formulas for the partial derivatives of inverse functions

Therefore I can say for and ,

where

Now I’ll show some examples on that.

#### Solved examples of partial derivatives of inverse functions

So here are my examples.

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

##### Example 1

According to Stroud and Booth (2011), “If and , determine . ”

##### Solution

Now here I have two functions and . And both of these functions are dependent on and .

So if I would have to find out or or or something similar, then there’s no problem.

Then I can use the method of the first-order partial derivative of functions with two variables.

But in here, I have to find out the value of and not .

So I’ll start with the Jacobians.

###### Step 1

(1)

(2)

First of all, I’ll differentiate equation (1) partially with respect to to get the value of as

Then I’ll differentiate equation (1) partially with respect to to get the value of as

Next, I’ll differentiate equation (2) partially with respect to to get the value of as

At the end, I’ll differentiate equation (2) partially with respect to to get the value of as

Now the Jacobian for the functions is

So in this example, it will be

Now I’ll evaluate the determinant to get the value of as

Next, I’ll simplify it to get

Now here is the common term. So I’ll take it out to get the value of as

(3)

As a next step, I’ll use the value of from equation (3) to get the partial derivatives of inverse functions.

###### Step 2

Thus, according to the formula mentioned above, the value of will be

Now I’ll simplify it. So it will be

Next, I’ll find the value of .

So it will be

Hence it gives

Now I’ll find the value of . So it will be

Next, I’ll simplify it. So it will be

Finally, I’ll find the value of . So it will be

Hence it gives

Therefore my conclusion is and are the answers to this example.

Now I’ll give another example on partial derivatives of inverse functions.

So here comes my next example.

##### Example 2

According to Stroud and Booth (2011), “If with and , determine expressions for and .”

##### Solution

Now in this example, I have three functions and . And all these three functions are dependent on and .

In order to get the values of and , I’ll start with the values of .

So here comes my first step.

First of all, I’ll find out the Jacobians.

Here the functions are

###### Step 1

First of all, I’ll differentiate the function partially with respect to to get the value of as

(4)

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

(5)

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

(6)

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

(7)

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

(8)

At the end, I’ll differentiate the function partially with respect to to get the value of as

(9)

Now the Jacobian for the functions is

Then I’ll evaluate the determinant to get the value of as

Next, I’ll simplify it to get

Now here is the common term. So I’ll take it out to get the value of as

As a next step, I’ll use this value of to get the partial derivatives of inverse functions.

###### Step 2

Thus, according to the standard formulae, the value of will be

So I’ll get the value of from equation (9) to find the value of as

(10)

Next, I’ll find the value of .

So it will be

Now I’ll get the value of from equation (8) to find the value of as

(11)

Next, I’ll find the value of .

So it will be

Now I’ll get the value of from equation (7) to find the value of as

(12)

Finally, I’ll find the value of .

So it will be

Now I’ll get the value of from equation (6) to find the value of as

(13)

As a next step, I’ll use the values of and from equations (10)-(13) to find out the values of and .

###### Step 3

According to the standard formula, the value of will be

Now I’ll substitute the values of from equation (4), from equation (5), from equation (10) and from equation (11) to get the value of as

So this gives the value of as

Also, as per the standard formula, the value of will be

Therefore, I’ll substitute the values of from equation (4), from equation (5), from equation (12) and from equation (13) to get the value of as

So this gives the value of as

Hence I can conclude that and where is the solution of this example.

Dear friends, this is the end of my today’s post on partial derivatives of inverse functions. 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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