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Implicit differentition: inverse functions,can someone please help me with my working

  1. Aug 3, 2012 #1
    1. The problem statement, all variables and given/known data

    I'm trying to solve the derivative of this equation: 1-(x*arcsin(x))/√(1-x^2 )
    I've straight away disregarded the 1 as it will be 0 so I'm left with -(x*arcsin(x))/√(1-x^2 ).

    3. The attempt at a solution

    I've applied the quotient rule and labelled U= -x*arcsin(x) and V=√(1-x^2 )
    I then got du/dx= (-x)/√(1-x^2 )-arcsin⁡(x) and dv/dx= -x/√(1-x^2 )

    I then used the quotient rule:
    [√(1-x^2 )*{(-x)/√(1-x^2 )-arcsin⁡(x)}]-[-x*arcsin(x)*-x/√(1-x^2 )] / [√(1-x^2 )]^2

    which then simplified to: [-x - arcsin⁡(x)*√(1-x^2 ) - (x^2)*arcsinx/√(1-x^2 )] / (1-x^2)

    I then multiplied top and bottom by √(1-x^2 ) and got:
    [-x√(1-x^2 ) - arcsinx*(1-x^2 ) - x^2*arcsinx] / (1-x^2 )*√(1-x^2 )

    Then I seperated everything to be individual and simplified:
    -x/(1-x^2) - arcsinx/√(1-x^2 ) - x^2*arcsinx/(1-x^2 )*√(1-x^2 )

    the real answer is meant to look like:
    -x/(1-x^2) -arcsinx/(1-x^2)^(3/2)

    I'm getting the first term but with the second term mine has an x^2 which I don't want and also I'm not seeing a (1-x^2)^(3/2) coming up anytime soon.

    It would be greatly appreciated if someone could look over my work for any mistakes and also if someone could point me in the direction to finish the equation.
    Thanks for your help in advance.
     
  2. jcsd
  3. Aug 3, 2012 #2
    Re: Implicit differentition: inverse functions,can someone please help me with my wor

    These are identical. Just collect and simplify the arcsin terms.
     
  4. Aug 3, 2012 #3
    Re: Implicit differentition: inverse functions,can someone please help me with my wor

    Ok I've cleared most of it up now but I have to ask one thing, how does - arcsin(x)*(1-x^2) - x^2*arcsin(x) = -arcsinx? I just dont understand the proccess of whats happening there.
     
  5. Aug 3, 2012 #4
    Re: Implicit differentition: inverse functions,can someone please help me with my wor

    Actually dont worry I spoke to soon, I missed a simple rule of expanding the brackets. Thanks heaps for your help.
     
  6. Aug 3, 2012 #5

    SammyS

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    Re: Implicit differentition: inverse functions,can someone please help me with my wor

    Why does the title mention implicit differentiation ?
     
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