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)
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