Differentiate inverse trigonometric functions. Hello friends, today I’ll show how to differentiate the inverse trigonometric functions. Have a look!!

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

**Differentiation of parametric functions**

**Logarithmic differentiation of functions**

**Differentiation of implicit functions**

**Differentiate the inverse trigonometric functions**

**Examples of how to differentiate the inverse trigonometric functions**

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 (2013)* “Differentiate .”

**Solution**

Let

(1)

And this is because if

First of all, I’ll differentiate both sides of the equation (1) with respect to . And I’ll use the * quotient rule of differentiation* for that together with the standard

*for that. So I get*

**formulas in differentiation of functions**

Then I’ll simplify it to get

(2)

Also, I know that . Hence I can replace in equation (2) with its value from equation (1). Therefore it becomes

So this means

Now I’ll simplify it to get

(3)

Next, I’ll substitute equation (3) to equation (2) to get

So this gives

Hence I can conclude that this is the answer to the given example. Now I’ll give another example.

**Example 2**

According to Stroud and Booth (2013)* “Differentiate .”

**Solution**

Let

First of all, I’ll differentiate both sides with respect to . And that means

Now in this example, I’ll use the * product rule of differentiation *together with the

*.*

**chain rule of differentiation**Thus I get

So this means

Next, I’ll simplify it a bit more to get

Then I’ll simplify it further to get

Hence I can conclude that is the answer to the given example. Now I’ll give another example.

**Example 3**

According to Stroud and Booth (2013)* “If ,

**Solution**

Now here the given function is

(4)

**(a)**

And I have to prove the relation .

First of all, I’ll differentiate equation (4) with respect to . Again I’ll use the quotient rule of differentiation like Example 1.

So that gives

Next, I’ll simplify it to get

Then I can say that

Again from equation (4), I know that . So I can say that

(5)

Hence I can conclude that I have proved the relation. And this is the answer to part (a) of the given example. Now I’ll do the part (b).

**(b)**

Now here I have to prove the relation .

So I’ll differentiate equation (5) with respect to . Again I’ll use the product rule of differentiation like Example 2. So that gives

Next, I’ll simplify it to get

Hence I can conclude that I have proved the relation. And this is the answer to part (b) of 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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