`y=sin^(-1)[sqrt(x-a x)-sqrt(a-a x)] then prove that 1/(2sqrtx sqrt(1-x))`

(sqrt(x))/(1 + sqrt(x))

(sin ^(-1) x )/( sqrt( 1 - x ^(2)) )

If y=sqrt(x+1)+sqrt(x-1) , prove that sqrt(x^(2)-1)(dy)/(dx)=1/2y

y = tan^(-1)((sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))),find dy/dx.

If sqrt(1-x^6)+sqrt(1-y^6)=a(x^3-y^3), then prove that (dy)/(dx)=(x^2)/(y^2)sqrt((1-y^6)/(1-x^6))

d/(dx)[cos^(-1)(xsqrt(x)-sqrt((1-x)(1-x^2)))]= 1/(sqrt(1-x^2))-1/(2sqrt(x-x^2)) (-1)/(sqrt(1-x^2))-1/(2sqrt(x-x^2)) 1/(sqrt(1-x^2))+1/(2sqrt(x-x^2)) 1/(sqrt(1-x^2)) 0 b. 1//4 c. -1//4 d. none of these

Differentiate w.r.t.x the function. cot^(-1)[(sqrt(1+sin x)+sqrt(1-sin x))/(sqrt(1+sin x)-sqrt(1-sin x))],0 lt x lt (pi)/(2) .

y=tan^(-1)(x/(1+sqrt(1-x^2)))

Find (dy)/(dx) in the following : y= sin^(-1) (2x sqrt(1-x^(2))), (1)/(sqrt(2)) lt x lt (1)/(sqrt(2)) .

y=tan^(-1)((sqrt(1+x^2)+sqrt(1-x^2))/(sqrt(1+x^2)-sqrt(1-x^2))),w h e r e -1