how to use the first principle to determine derivative of the function f(x)=square root(7x+5)

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 [7(x+h)+5] -
sqrt(7x+5)}/h

We'll remove the brackets from
radicand:

lim [sqrt (7x+7h+5) -
sqrt(7x+5)]/h

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

lim [sqrt (7x+7h+5) - sqrt(7x+5)][sqrt
(7x+7h+5)+sqrt(7x+5)]/h*[sqrt (7x+7h+5)+sqrt(7x+5)]

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

lim [(7x+7h+5) -
(7x+5)]/h*[sqrt (7x+7h+5)+sqrt(7x+5)]

We'll eliminate like terms
form numerator:

lim 7h/h*[sqrt
(7x+7h+5)+sqrt(7x+5)]

We'll simplify and we'll
get:

lim 7/[sqrt
(7x+7h+5)+sqrt(7x+5)]

We'll substitute h by
0:

lim 7/[sqrt (7x+7h+5)+sqrt(7x+5)] =
7/[sqrt(7x+5)+sqrt(7x+5)]

We'll combine like terms from
denominator:

f'(x)=7/2sqrt(7x+5)