What is the reason for the derivative of arcsin(x) not being -1/sqrt(1-x^2)?

[Petrus]

Hello MHB,
derivate $$\sin^{-1}(x)$$So I use the derivate formula for invers and get
$$\frac{1}{\cos(\sin^{-1}(x))}$$
and Then draw it and get $$\frac{1}{\sqrt{1-x^2}}$$
but there is a reason WHY it can't be $$-\frac{1}{\sqrt{1-x^2}}$$ and I did not understand it, I did not get it.

Regards,
$$|\pi\rangle$$

 

[I like Serena]

Re: Derivate of arcsin(x)

Hello MHB,
derivate $$\sin^{-1}(x)$$So I use the derivate formula for invers and get
$$\frac{1}{\cos(\sin^{-1}(x))}$$
and Then draw it and get $$\frac{1}{\sqrt{1-x^2}}$$
but there is a reason WHY it can't be $$-\frac{1}{\sqrt{1-x^2}}$$ and I did not understand it, I did not get it.

Regards,
$$|\pi\rangle$$

The $\arcsin$ is defined to have a range of $-\pi/2$ to $+\pi/2$.
With this definition the derivative is always positive.
You can also choose the range to be different, making the derivative negative, but then it's not an $\arcsin$ anymore. Then you have a different inverse for the sine.