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

[e_brock123]

Homework Statement

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 ).

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 separated 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.

 

[voko]

Then I separated 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)

These are identical. Just collect and simplify the arcsin terms.

 

[e_brock123]

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 don't understand the process of what's happening there.

 

[e_brock123]

Actually don't worry I spoke to soon, I missed a simple rule of expanding the brackets. Thanks heaps for your help.

 

[SammyS]

Actually don't worry I spoke to soon, I missed a simple rule of expanding the brackets. Thanks heaps for your help.

Why does the title mention implicit differentiation ?