You did not expand nor rationalize correctly.
A fit conjugate would be . By your way, you should've got:
Can anyone see where I'm going wrong here...
So differentiate from 1st principles.
f(x) = sqrt(x)+1
f'(x) = (sqrt(x+h)+1)-(sqrt(x)+1) / h
f'(x) = x+h+1-x-1 / h(sqrt(x+h)+1)+(sqrt(x)+1)
So cancelling terms i get,
f'(x) = 1 / (sqrt(x+h)+1)+(sqrt(x)+1)
f'(x) = 1 / 2sqrt(x) + 2
This is where I'm confused, shouldn't it be
f'(x) = 1 / 2sqrt(x)