There are a few different versions of the limit definition of a derivative. For this answer, I'll opt for the following definition:
#f'(x) = lim_ {h->0} (f(x+h)-f(x))/h#
Thus, we continue:
#f'(x) = lim_ {h->0} (sqrt((x+h)+1) - sqrt(x+1))/h#
# = lim_ {h->0} (sqrt((x+h)+1) - sqrt(x+1))/h * (sqrt((x+h)+1) + sqrt(x+1))/(sqrt((x+h)+1) + sqrt(x+1)) #
# = lim_ {h->0} ((sqrt((x+h)+1))^2 - (sqrt(x+1))^2)/(h * (sqrt((x+h)+1) + sqrt(x+1)) #
# = lim_ {h->0} (x+h+1 - (x+1))/(h * (sqrt((x+h)+1) + sqrt(x+1)) #
# = lim_ {h->0} (x+h+1 - x - 1)/(h * (sqrt((x+h)+1) + sqrt(x+1)) #
# = lim_ {h->0} cancel(h)/(cancel(h) * (sqrt((x+h)+1) + sqrt(x+1)) #
# = lim_ {h->0} 1/(sqrt((x+h)+1) + sqrt(x+1)) #
# = 1/(sqrt(x+1) + sqrt(x+1)) = 1/(2sqrt(x+1)) #