Derivative of a fraction inside a radical

[ehh]

f(z) = sq. rt of z-1 / z+1 --- both numerator and denominator are inside the radical.

I can write it as (z-1)^1/2 over (z+1)^1/2, right? If I simplify it using derivative of a quotient. Should I simplify (z-1)^1/2 and (z+1)^1/2 as whole numbers and multiply them to other terms, including adding the exponents? The teach said I couldn't because the one-halves are actually square roots so I can't count them as exponents. Help?

 

[UltrafastPED]

√x = x^1/2; when taking derivatives of radicals you should always convert them to exponents.

 

[HallsofIvy]

You can differentiate $$\sqrt[2]{\frac{z- 1}{z+ 1}$$ by treating it as $$\frac{(z- 1)^{1/2}}{(z+ 1)^{1/2}}$$ using the quotient theorem, the chain rule, and the power rule, in that order. Or think of it as $$\left(\frac{z-1}{z+1}\right)^{1/2}$$ using the same rule in a different order. Or think of it as $$(z- 1)^{1/2}(z+ 1)^{1/2}$$ and use the product rule rather than the quotient rule.