# How do you differentiate #y=arcsin(x)#?

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I don't even know where to start.

I watched a video on You Tube of how to do it, but it just didn't make any sense to me. Hope someone can help.

Thanks

I don't even know where to start.

I watched a video on You Tube of how to do it, but it just didn't make any sense to me. Hope someone can help.

Thanks

##### 3 Answers

See below.

#### Explanation:

Before we proceed we need to understand just what it is we are looking for. Remember that:

This can be expressed as:

Using

We need to differentiate in respect of

Remembering that:

From above

Substitutng in

This is a little awkward, and it would be easier if we could express this in a different way.

Using the Pythagorean identity:

Using positive root: **( see below)**

From above

Hence:

Reason for using positive root of

This is because **only has an inverse** if we restrict the domain to:

In

This range is in the **I** and **IV** quadrants, where the cosine ratios are positive.

Here's an alternative solution

#### Explanation:

This troubled me as well when I was first learning these kinds of derivatives. Here's how I learned how to tackle this type of problem:

Similarly to what was said below,

When trying to take the derivative of

**It is at this point where I would create a right triangle showing this**

Recall:

In this case

We can solve for the missing side using **This will be useful later!**

Now we'll differentiate implicitly (W.R.T

Divide

Now we'll have to rewrite in terms of

Well based on the triangle we made above:

So

The neat thing about this is that this is useful for the other inverse functions like

Create a triangle:

Differentiate both sides implicitly (W.R.T

Divide both sides by

Rewrite in terms of

Since

Then

So

Also if you have something like

So...

Differentiate implicitly (W.R.T

Since

Then

So

I hope I explained this well!

#### Explanation:

There is another way to do it. We have a formula for calculating the derivatives of inverse functions:

Here

So

Using the arguments given in the other answers, we know that