Calculus of a Single Variable, Chapter 2, 2.1, Section 2.1, Problem 23

By limit process, the derivative of a function f(x) is :-
f'(x) = lim h --> 0 [{f(x+h) - f(x)}/h]
Now, the given function is :-
f(x) = sqrt(x+4)
THus, f'(x) = lim h --> 0 [{f(x+h) - f(x)}/h]
or, f'(x) = lim h ---> 0 [{(sqrt(x + h+ 4)) - sqrt(x+4)}/h]
rationalizing the numerator we get
f'(x) = lim h ---> 0
[{(sqrt(x + h+ 4)) - sqrt(x+4)}*{(sqrt(x + h+ 4)) + sqrt(x+4)}/{h*{(sqrt(x + h+ 4)) + sqrt(x+4)}}]
or, f'(x) = lim h ---> 0 [{(x+h+4) - (x+4)}/{h*{(sqrt(x + h+ 4)) + sqrt(x+4)}}]
or, f'(x) = lim h ---> 0 [h/{h*{(sqrt(x + h+ 4)) + sqrt(x+4)}}]
or, f'(x) = lim h ---> 0 [1/{1{(sqrt(x + h+ 4)) + sqrt(x+4)}}]
putting the value of h = 0 in the above expression we get;
f'(x) = 1/{2*sqrt(x+4)}