Using the definition of derivative find the derivative of y=(2x+4)^1/2

First, we'll express the first principle of finding the
derivative of a given function:

lim [f(x+h) - f(x)]/h, for
h->0

We'll apply the principle to the given
polynomial:

lim {sqrt [2(x+h)+4] -
sqrt(2x+4)}/h

The next step is to remove the brackets under
the square root:

lim [sqrt (2x+2h+4) -
sqrt(2x+4)]/h

We'll remove multiply both, numerator and
denominator, by the conjugate of numerator:

lim [sqrt
(2x+2h+4) - sqrt(2x+4)][sqrt (2x+2h+4)+sqrt(2x+4)]/h*[sqrt
(2x+2h+4)+sqrt(2x+4)]

We'll substitute the numerator by the
difference of squares:

lim [(2x+2h+4) - (2x+4)]/h*[sqrt
(2x+2h+4)+sqrt(2x+4)]

We'll eliminate like terms form
numerator:

lim 2h/h*[sqrt
(2x+2h+4)+sqrt(2x+4)]

We'll simplify and we'll
get:

lim 2/[sqrt
(2x+2h+4)+sqrt(2x+4)]

We'll substitute h by
0:

lim 2/[sqrt (2x+2h+4)+sqrt(2x+4)] =
2/[sqrt(2x+4)+sqrt(2x+4)]

We'll combine like terms from
denominator:

f'(x)=1/sqrt(2x+4)