What is the derivative of x−−√ using first principles?

Answer 1

The function is not clear here. If I assume you mean f(x)=x^(1/2), then to find its derivative using first principles just means to use the definition f '(x) = lim as h-->0 of [f(x+h)-f(x)] / h. For simple square roots, you can evaluate this limit by rationalizing the numerator.