Hello friends, today I will talk about the differentiation of implicit functions. Have a look.
Differentiation of implicit functions
Now the first question is: what is an implicit function?
Any function which looks like but not the more common is an implicit function.
For example, is an implicit function.
But is not an implicit function. This is called an explicit function.
In any implicit function, it is not possible to separate the dependent variable from the independent one.
Now I will solve an example of the differentiation of an implicit function.
Example of the differentiation of implicit functions
Disclaimer: 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 example.
According to Stroud and Booth (2013)* “Find when ”
Here the given function is
Now to get the value of , I’ll differentiate the equation (1).
Here again, I’ll use the standard formulas in differentiation.
See also: Standard formulas in differentiation
Differentiating equation (1) throughout with respect to , I get
Now here both and are single functions.
So it’s quite straightforward to differentiate them.
But the third term is a product of two functions.
For that, I’ll use product rule of differentiation.
You can also check out the product rule of differentiation
Thus it will be
My next step will be to rearrange the terms.
Now I’ll rearrange the terms to get
Here I can see that ‘3’ is common to each term.
Next, I’ll cancel ‘3’ from each term to get
Now I’ll separate terms with
Thus it will be like
At the end, I’ll simply rearrange the terms to get the final value of as
Thus the value of is
This is the answer to the given problem.
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!!