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.
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
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
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.