Hello friends, today I will talk about the differentiation of implicit functions. Have a look.

If you’re looking for more in the differentiation of functions, do check out:

**Differentiation of parametric functions**

**Differentiation of logarithmic functions**

**How to differentiate inverse trigonometric functions**

Differentiation of implicit functions_compressed

### 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.

**Examples 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.

**Example 1**

According to Stroud and Booth (2013)* “Find when

**Solution**

Here the given function is

(1)

Now to get the value of

Here again, I’ll use the standard * formulas in differentiation*.

**Step 1**

Differentiating equation (1) throughout with respect to

Now here both

So it’s quite straightforward to differentiate them.

But the third term

For that, I’ll use the * product rule of differentiation*.

Thus it will be

My next step will be to rearrange the terms.

**Step 2**

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

Thus the value of

This is the answer to the given example. Now I’ll give another example.

**Example 2**

According to Stroud and Booth (2013)* “If

**Solution**

Now here the given function is

(2)

And I have to prove the relation

**Step 1**

Therefore, I’ll differentiate both sides of equation (2) with respect to

Now that gives

(3)

Then I’ll differentiate both sides of equation (3) with respect to

(4)

Now I’ll prove the relation

**Step 2**

As I can see in equation (4),

Hence I’ll multiply equation (4) throughout with

(5)

Now from equation (3), I can say that

So equation (5) becomes

Next, I’ll simplify it to get

which means

And I can also rewrite it as

Hence I have proved the relation and this is the answer to the given 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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