Question

Why do we use the Chain Rule to find the derivative of the inverse sine?

`arcsinx` or `sin^-1x` is not a composite function, so why do we apply the Chain Rule to find the derivative?

Answer 1

Interesting question...

Explanation:

First, note here that `d/dx[sin^-1(x)]=1/(sqrt(1-x^2))`

In the derivative, the domain is `(-1,1)`.

Essentially, we limit the range of `arcsin(x)` to `(-pi/2,pi/2)`.

This way, it is a function.

(By the way, we use a capital A for arcsin to represent it as a function.)

Also, if you look at the graph of the derivative, the function "explodes" as it reaches `-1` and `1`. This is because in the graph of the original function, the wave becomes straighter and straighter vertically as we approach `1` and `-1`.