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

If y=(x)/(sqrt(1+x^(2))) , prove that, x^(3)(dy)/(dx)=y^(3) .

If y sqrt(x^(2)+1)=log(sqrt(x^(2)+1)-x) , show that, (x^(2)+1)(dy)/(dx)+xy+1=0

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

If y=sqrt((1-x)/(1+x)) , prove that , (1-x^(2)) (dy)/(dx)+y=0

If y= (x)/(2) sqrt(x^(2)-a^(2))-(a^(2))/(2) log (x+sqrt(x^(2)-a^(2))) , show that, (dy)/(dx)=sqrt(x^(2)-a^(2))

If y=(sqrt(x))^((sqrt(x))^((sqrt(x))^(...oo) , show that, x (dy)/(dx)=(y^(2))/(2-y log x).

If y="tan"^(-1) (sqrt(1+x^(2))-sqrt(1-x^(2)))/(sqrt(1+x^(2))+sqrt(1-x^(2))) show that, (dy)/(dx)=(x)/(sqrt(1-x^(4)))

If y^(x)=e^(y-x)" prove that, " (dy)/(dx)=((logey)^(2))/(log y) .

If y = sqrt(x/(m))+sqrt(m/(x)) show that 2xy (dy)/(dx) = (x)/(m) - (m)/(x) *

If x^(logy)=logx , prove that, (x)/(y).(dy)/(dx)=(1-logx log y)/((log x)^(2))