# What is the derivative of arcsin[x^(1/2)]?

To find the derivative we will need to use the Regola di derivazione

dy/dx=dy/(du)*(du)/(dx)

Vogliamo trovare

d/(dx)(arcsin(x^(1/2)))

Dopo l' regola di derivazione lasciamo u=x^(1/2)

Deriving u we get

(du)/(dx)=1/2*x^(-1/2)=1/(2sqrt(x))

Now we substitute u in place of x in the original equation and derive to find dy/(du)

y=arcsin(u)

(dy)/(du)=1/(sqrt(1-u^2)

Now we substitute these derived values into the chain rule to
Find dy/(dx)

dy/dx=dy/(du)*(du)/(dx)

dy/dx=1/(sqrt(1-u^2))*1/(2sqrt(x))

Substitute x back into the equation to get the derivative in terms of x only and simplify

u=x^(1/2)

dy/dx=1/(sqrt(1-(x^(1/2))^2))*1/(2sqrt(x))

dy/(dx)=1/(sqrt(1-x))*1/(2sqrt(x))

dy/(dx)=1/(2sqrt(x)*sqrt(1-x))

dy/(dx)=1/(2sqrt(x-x^2))