Question

`y=sin^-1[x(sqrt(x-1)) - sqrtx sqrt(1-x^2)]`, find `dy/dx`?

Answer 1

`(dy)/(dx)=(sqrt(x-1)+x/(2sqrt(x-1))+(2xsqrtx)/(2sqrt(1-x^2))-sqrt(1-x^2)/(2sqrtx))/sqrt(1-x+x^2+2xsqrt(x^2-x^4-x+x^3)`

Explanation:

We can use chain rule here.

Let `y=sin^(-1)(g(x))`, where `g(x)=xsqrt(x-1)-sqrtxsqrt(1-x^2)`

then `(dy)/(dx)=(dy)/(dg(x))xx(dg(x))/dx`

Now `(dy)/(dg(x))=1/sqrt(1-g(x)^2)`

as `g(x)^2=(xsqrt(x-1)-sqrtxsqrt(1-x^2))^2`

= `x^2(x-1)+x(1-x^2)-2xsqrtxsqrt((x-1)(1-x^2))`

= `x^3-x^2+x-x^3-2xsqrt(x^2-x^4-x+x^3)`

= `x-x^2-2xsqrt(x^2-x^4-x+x^3)`

and `(dg)/(dx)=sqrt(x-1)+x*1/(2sqrt(x-1))-sqrtx*(-2x)/(2sqrt(1-x^2))-1/(2sqrtx)sqrt(1-x^2)`

= `sqrt(x-1)+x/(2sqrt(x-1))+(2xsqrtx)/(2sqrt(1-x^2))-sqrt(1-x^2)/(2sqrtx)`

Hence `(dy)/(dx)=(sqrt(x-1)+x/(2sqrt(x-1))+(2xsqrtx)/(2sqrt(1-x^2))-sqrt(1-x^2)/(2sqrtx))/sqrt(1-x+x^2+2xsqrt(x^2-x^4-x+x^3)`