Verified answer

Let f(x) = sqrt(x - 1).

Then the definition of the derivative says:

f'(x) = lim as h -> 0 of [f(x + h) - f(x)] / h.

Then:

[sqrt((x + h) - 1) - sqrt(x - 1)] / h.

Multiply both sides of the fraction by sqrt(x + h - 1) + sqrt(x - 1), giving us:

[sqrt(x + h - 1)^2 - sqrt(x - 1)^2] / [ sqrt(x + h - 1) + sqrt(x - 1) ], which comes out to:

[x + h - 1 - (x - 1)] / [h * sqrt(x + h - 1) + sqrt(x - 1)], which is:

h / [h * sqrt(x + h - 1) + sqrt(x - 1)], and we divide h out of both sides, giving us:

1 / [sqrt(x + h - 1) + sqrt(x - 1)]. As h gets very small, that is:

1 / 2(sqrt(x - 1)), and you're done.

Hope this helps.