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 ?

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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 ?

OCW Scholar - Single Variable Calculus
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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  • 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
I see , thanks a lot , makes perfect sense

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