Help using the definition of a derivative to solve for 1/x^1/2?

Question

I am trying to do some problems for calc review, and everytime I attempt this problem, I end up getting stuck, anyone care to show me how this problem is done?

f(x)= 1/x^(1/2) assuming x>0.

I need to use the definition f'(x)=lim h->0 (f(x+h)-f(x))/h

Answer 1

f(x+h) - f(x) = 1/√(x+h) - 1/√x
= [√x - √(x+h)]/[√(x+h) √x]

Standard trick: multiply numerator and denominator by
√x + √(x+h):

f(x+h) - f(x) = [√x - √(x+h)][√x + √(x+h)]/[(√(x+h) √x)(√x + √(x+h))]

= [x - (x+h)]/[(√(x+h) √x)(√x + √(x+h))]

= -h/[(√(x+h) √x)(√x + √(x+h))]

So

[f(x+h) - f(x)]/h = -1/[(√(x+h) √x)(√x + √(x+h))] for h≠0

Now the limit as h → 0 is easy to evaluate

Answer 2

???cut back??[ ?(a million + 2(x + h)) ?– ??(a million + 2x) ] ? h?=?0 ? 0 ?h???0 multiply with the help of: [ ?(a million + 2(x + h))?+??(a million + 2x) ]? ? ?[ ?(a million + 2(x + h))?+??(a million + 2x) ] ???...?then boost and simplify the numerator?... ???cut back??2h? ? ?h?•?[ ?(a million + 2(x + h))?+??(a million + 2x) ] ?h???0 ???cut back??2? ? ?[ ?(a million + 2(x + h))?+??(a million + 2x) ]? = ?h???0 ?????=?2? ? ?[ ?(a million + 2x)?+??(a million + 2x) ] ?????=?a million? ? ??(a million + 2x)