Question

Find dy/dx :
y=sin^-1[ x{ 1-x }^1/2 - { x }^1/2 {1-( x )^2}^1/2]

Answer 1

I'll rewrite it,
\[y=\arcsin (x\sqrt{1-x}-\sqrt{x}\sqrt{1-x^2})\]
is this the y you wrote?

Answer 2

Is this the case, then, the derivative is,
\[dy/dx=\frac{-\frac{\sqrt{1-x^2}}{2 \sqrt{x}}+\frac{x^{3/2}}{\sqrt{1-x^2}}-\frac{x}{2 \sqrt{1-x}}+\sqrt{1-x}}{\sqrt{1-\left(\sqrt{1-x} x-\sqrt{x} \sqrt{1-x^2}\right)^2}}\]If you need step by step, tell me.

Answer 3

Yes,please!!

Answer 4

You need to apply the derivative of the arcsin, and chain rule
\[(\arcsin (f(x)))'=\frac{f'(x)}{\sqrt{1-(f(x))^2}}\]
And of course, the derivative of the square root,
\[(\sqrt{f(x)})'=\frac{f'(x)}{2\sqrt{f(x)}}\]
With these tools, you can follow the process.

Answer 5

Got it!!

Answer 6

Ok. ;)

Answer 7

And thanks.. ;)

Answer 8

You're welcome.

Answer 9

One more question!!!

Is there a way of substituting some value in place of x and solving the problem ?

Answer 10

Yes, of course, with the final formula of the derivative, plug the value you want there and you will have the solution. Although, as you can see, it will be a little clumsy to plug in the value in this big formula.

Answer 11

Hmmm..I see...