Put z = x + h.

Then:

f'(x) = lim(h -> 0)( f(x + h) – f(x) ) / h

f'(x) = lim(h -> 0)( (2(s + h) + 1)^(1/2) – (2s + 1)^(1/2) ) / h

Multiply top and bottom by

(2(s + h) + 1)^(1/2) – (2s + 1)^(1/2)

to rationalise the numerator:

f'(x) = lim(h -> 0)( { [(2(s + h) + 1)^(1/2) – (2s + 1)^(1/2) ]

* [(2(s + h) + 1)^(1/2) + (2s + 1)^(1/2) ] }

/ h [ (2(s + h) + 1)^(1/2) + (2s + 1)^(1/2) ]

The numerator is now of the form (a – b)(a + b), which is equal to a^2 – b^2.

Hence:

f'(x) = lim(h -> 0) [ 2(s + h) + 1 – (2s + 1) ]

/ [ h (2(s + h) + 1)^(1/2) + (2s + 1)^(1/2) ]

= lim(h -> 0) ( 2h / h [ (2(s + h) + 1)^(1/2) + (2s + 1)^(1/2) ] )

= lim(h -> 0) ( 2 / [ (2(s + h) + 1)^(1/2) + (2s + 1)^(1/2) ] )

= 2 / [ 2(2s + 1)^(1/2) ]

= 1 / (2s + 1)^(1/2).