Question

use chain rule to differentiate y=(square root x -1)^3

Answer 1

start by converting roots to exponents

But it not clear to me whether you mean sqrt(x-1) or sqrt(x) - 1. I'm just guessing you mean sqrt(x-1), so in all it is ((x-1)^(1/2))^3

f(x) = x^3
g(x) = (x-1)^(1/2)

now actually there is a third function h(x) = x -1, but as you can see its derivative is 1, so many people will unconsciously realize that and not include it explicitly.

The derivative of f(g(x)) is f'g(x)g'(x), that's the chain rule.

so

d/dx [((x -1)^(1/2))^3] = [3((x-1)^(1/2))^2][ (1/2)(x-1)^(-1/2)]

simplify to taste.

If on the other hand you did mean (x^(1/2) - 1)^3

f(x) = x^3
g(x) = x - 1
h(x) = x^(1/2)

and the chain rule is

d/dx f(g(h(x))) = f'(g(h(x)))g'(h(x))h'(x) and of course you can extend this as necessary in cases where you have even more nested functions.