How do you find the derivative of ##y=arcsin(sqrt(x))##?

Here you want* So , let ##sqrtx## =## z##
##dy/(dx)## =##d/(dz## x##dz/dx##===>
It’s better to take ##d/(dz## & ## dz/dx## separately .
##d/(dz##= ##d/(dsqrtx##.##arcsin (sqrtx)##.
##d/(dz##= ##1/sqrt(1-(sqrtx)^2)##= ##1/sqrt(1-x)## ——(1)
##dz/dx##= ##d/dx## .##(sqrtx)## =##1/(2sqrtx## ——(2)
Then multiply (1) &(2),
So finally you get ##dy/(dx)##=##1/sqrt(1-x)####1/(2sqrtx##
##dy/(dx)##=##1/(2sqrt(x(1-x))## ; where ##y## = ##arcsinsqrtx## to use the chain rule.
If you want to know how to differentiate ##arctansqrtx## check http://socratic.org/questions/how-do-you-find-the-derivative-of-y-arctan-sqrt-x