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  • 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 ?

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  1. phi
    • one year ago
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    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
    • one year ago
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    I see , thanks a lot , makes perfect sense

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