Calculating the Derivative of Inverse Trig Functions

The formulas for the derivative of inverse trig functions are one of those useful formulas that you sometimes need, but that you don't use often enough to memorize. That's why I think it's worth your time to learn how to deduce them by yourself.

In this page I'm going to show you an easy and quick method to prove to yourself these formulas in less than a minute. Knowing this, you won't have to worry about forgetting these formulas.

So, let's first present them. These are the most important formulas

Let's now deduce these quickly and easily. The only tool we'll need is the chain rule

Derivative of arcsin(x)

Let's begin with inverse sin function. By definition, arsin(x) is the function whose sine is x. That is, the function is defined by the equation

The basic idea is to take the derivative of both sides of the equation, applying the chain rule on the left side. To make things clear, we'll introduce the variable y

That is a simple equation we have up there. Now we take the derivative of both sides, reminding ourselves that y is a function of x

Now we solve for y', which is what we are trying to calculate

We already have an expression for the derivative. Now, we want to simplify it. To do that, we write it in terms of the original trigonometric function, in this case sin. We can write

And now, we have that

So, we finally obtain the first formula up there

Derivative of arccos(x)

The idea is the same. We start with the defining equation

We introduce a variable y to make the manipulations easier

Now we take the derivative of both sides, considering that y is a function of x

Now we solve for y'

We need to write this in terms of the original trigonometric function, cos

But we have that

Replacing that, we get to the formula

Derivative of arctan(x)

This one follows the same process, but in this case we need to use a more sophisticated trig identity. As before, we begin with the defining equation

Now we take the derivative of both sides

We need to write this equation in terms of the original trigonometric function, the tangent function. Here is where we need the following trig identity, which is easily proved

Also, we have that

Then, the formula for the derivative is

That's it. If you have any question or doubt about this topic, or just want to say hi, leave me a comment below.

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