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    What's the quickest way to find the derivative of f(x) = {\sin ^{ - 1}}\frac{x}{{\sqrt {{a^2} + {x^2}} }} ?

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    (Original post by BenjaminKyle)
    What's the quickest way to find the derivative of f(x) = {\sin ^{ - 1}}\frac{x}{{\sqrt {{a^2} + {x^2}} }}dx ?

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    Could you clarify what needs differentiating please?
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    Use the chain rule (twice) and the quotient rule.
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    ?

    1. first set u = x/(sqr(a^2+x^2)).
    2. then the differentiation of arcsinu is (du/dx)/sqrt of (1-u^2)
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    (Original post by BenjaminKyle)
    What's the quickest way to find the derivative of f(x) = {\sin ^{ - 1}}\frac{x}{{\sqrt {{a^2} + {x^2}} }}dx ?

    Thanks
    (Original post by JBrioso)
    ?

    1. first set u = x/(sqr(a^2+x^2)).
    2. then the differentiation of arcsinu is (du/dx)/sqrt of (1-u^2)
    I'd let

    y=f(x)

    so that

    \displaystyle \sin y = \dfrac{x}{\sqrt{x^2 + a^2}}

    Differentiate the LHS implicitly and it's a quick job with the quotient rule on the RHS.

    use

    \displaystyle \cos y = \sqrt{1-\sin^2 y}

    and you're done.
 
 
 
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