Question

I have trouble interpreting an answer to a question.
The question was: find the derivative of Arcsin (sqrt (1-x^2))

My answer was -1/sqrt (1-x^2). When I checked it, it said -1/sqrt (1-x^2) for x>0 , and +1/sqrt (1-x^2) for x < 0

Could someone please explain why is this so ?

Answer 1

It is a subtle issue.
\[ \frac{d}{dx} \sin^{-1}\left( \sqrt{1-x^2}\right) = \frac{1}{\sqrt{1-(1-x^2)}}\cdot \frac{-x}{\sqrt{1-x^2}}\]
the first term simplifies to \[\frac{1}{\sqrt{x^2}} = \frac{1}{x}\]
but notice that if x were originally negative, when we take the principal square root of x^2 we get a positive value. thus it is better to say
\[\frac{1}{\sqrt{x^2}} = \frac{1}{|x|}\]
and the derivative is
\[ \frac{x}{|x|}\cdot \frac{-1}{\sqrt{1-x^2}}\]
if x is positive, we get the expected result, but when x is negative , x/|x| = -1
and -1*-1 gives us the +1

Answer 2

I see , thanks a lot , makes perfect sense