Let us consider the function where is any arbitrary function of x and is any arbitrary nonzero constant.
We now proceed to determine the derivative formula. We first note that
We now differentiate both sides with respect to x to get:
Note that here, I let . Thus, we now right
Now, note that the range of is . Thus, is positive over such values, and thus we can take the square root of to get .
But this implies that since . Therefore, the derivative formula becomes the following:
But we continue to rewrite this as follows, using the fact that :
When you use this formula, keep in mind that can be any constant! So going back to your remark about taking the derivative of , note that is just another constant (think of it like ). So when you use the derivative formula, you always square the constant (as we already proved). The fact that we're working with an arbitrary constant in the formula gives us this flexibility. In short, whenever we have some constant appearing within the inverse sine function, it always ends up getting squared in its derivative.
I'm a little too tired to discuss the issue about arbitrary constant vs variable, so I'll leave it for someone else who wants to address that.
I hope this helps clarifies things.