find derivative using limit definition:

Question

f(x) = x - sqrt(x)

so f'(x) =
lim
h->0 [f(x+h) - f(x)]/h

but I keep trying to solve by multiplying by the conjugate but I can't figure it out..there's nothing that can be cancelled or anything and I can't get the derivative

sorry this is a repost, but i messed up my other one and i really need help on this

Answer 1

The key to this is the binomial theorem:

(x+h)^n = x^n + n*x^(n-1)*h + n(n-1)/2 * x^(n-2) * h^2 + ...

So, we find ourselves with

(h + sqrt(x) - sqrt(x+h))/h
= (h + x^(1/2) - (x^(1/2) + 1/2 * x^(-1/2)*h + <higher powers of h>)/h
= 1 - 1/2 * x^(-1/2)
(all terms with h go to zero)