From the title of your post I'm guessing you want to find f'(x). Your simplification isn't quite correct. Don't you mean ƒ(x) = (1-sin(x))^(1/2)? Fractional powers are treated the same as integer powers when it comes to differentiation. I presume you know how to differentiate y = x3. Use the exact same methodology to differentiate y = x1/2.I'm kind of confused about how to approach a function with the chain rule.
For example in the equation ƒ(x) = sqrt(1-sin(x)) I know i simplify it to ƒ(x) = 1-sin(x)^(1/2) but I'm lost from there.