Derivative of arcsin(x)

Petrus

Well-known member
Hello MHB,
derivate $$\displaystyle \sin^{-1}(x)$$

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

Regards,
$$\displaystyle |\pi\rangle$$

Klaas van Aarsen

MHB Seeker
Staff member
Re: Derivate of arcsin(x)

Hello MHB,
derivate $$\displaystyle \sin^{-1}(x)$$

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

Regards,
$$\displaystyle |\pi\rangle$$
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.