If `y=2tan^(-1) sqrt((x-a)/(b-x)), a lt x lt b`, show that, `((dy)/(dx))^(2)+(1)/((x-a)(x-b))=0`

If y=(sin^(-1) x)/(sqrt(1-x^2)) , then show that (1-x^2)(d^2y)/(dx^2)-3x(dy)/(dx)-y=0 .

If y= tan^(-1)((a cos x-b sin x)/(b cos x+a sin x))" show that, " (dy)/(dx)= -1 .

If tan y=(tan x+sec x-1)/(tan x-sec x+1), " show that, " (dy)/(dx)=(1)/(2) .

If y=tan^(-1)sqrt((x+1)/(x-1)),t h e n(dy)/(dx)i s (a) (-1)/(2|x|sqrt(x^2-1)) (b) (-1)/(2xsqrt(x^2-1)) (c) 1/(2xsqrt(x^2-1)) (d) none of these

"If "y=(2)/(sqrt(a^(2)-b^(2))){tan^(-1)(sqrt((a-b)/(a+b))tan""(x)/(2))}," then show that "(d^(2)y)/(dx^(2))=(b sin x)/((a+b cos x)^(2)).

If xsqrt(1+y)+ysqrt(1+x)=0," for ", -1 lt x lt 1 , prove that (dy)/(dx)= -(1)/((1+x)^(2)) .

If y=tan^-1((sqrt(1+x^2)-1)/x) , then dy/dx at x = 0 is

If 0 lt x lt pi/2 , show that, cos x gt 1-x^(2)/(2!)

If y=sqrt((a-x)(x-b))-(a-b)tan^-1sqrt((a-x)/(x-b)) ,show that dy/dx=sqrt((a-x)/(x-b))

y=tan^(-1) ((sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x)) show that dy/dx=1/(2sqrt(1-x^2))