Differentiation Problem (3)

If y = $\sin^{-1} \left[x\sqrt{1-x} -\sqrt{x} \sqrt{1-x^2} \right] $ and $ 0

Question

Differentiation Problem (3)

If y = $\sin^{-1} \left[x\sqrt{1-x} -\sqrt{x} \sqrt{1-x^2} \right] $ and $ 0 < x < 1$ , then find $\cfrac{dy}{dx}$

mathslover · Answer

@Miracrown @ganeshie8 .

anonymous · Answer

take x = sin theta

ganeshie8 · Answer

implicit differentiation may simplify the work a bit

mathslover · Answer

Ganeshie8 , how to use implicit differentiation here? and what is it?

mathslover · Answer

(*** My net is fluctuating... I may not give fast responses... So, please expect some late responses from my side. I'm sorry! **** )

ganeshie8 · Answer

lets wait :) looks @gemini208012  has more simpler method...

mathslover · Answer

Okay.

anonymous · Answer

wait a sec I know how to do it.

anonymous · Answer

x = sin theta
theta = sin ^-1 x

and root x = sin phi
phi = sin ^-1 root x
so,
y = sin ^-1 [sin theta .sqrt{1 - sin^2 phi}  - sin phi .sqrt {1 - sin^2 theta}

ganeshie8 · Answer

Nice :)

anonymous · Answer

which will give you 
y = sin^-1 x - sin ^-1 root x

anonymous · Answer

@ganeshie8 : thank for the medal.
@mathslover  did you get it ?

mathslover · Answer

Not sure whether I got it or not... As I am still trying to understand what you did above. But, I appreciate that, and when I complete reading that, I will let you know if I have some doubts or not.

mathslover · Answer

Yep. I got it completely @gemini208012 - Thanks a lot.

anonymous · Answer

Happy to be of some help.