## anonymous one year ago 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 ?

1. phi

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

2. anonymous

I see , thanks a lot , makes perfect sense