Question #75a4b

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Answer 1 · 2017-06-27T10:05:03

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The first principle defines the derivative of a function $f$ $f$ as such:

$f'=lim_(h->0)(f(x+h)-f(x))/(h)$

We let $f=1/sqrt(x+5)$
Then
$f'=lim_(h->0)(1/sqrt(x+h+5)-1/sqrt(x+5))/(h)$

$=lim_(h->0)1/h*(1/sqrt(x+h+5)-1/sqrt(x+5))$

$=lim_(h->0)1/h*((sqrt(x+5)-sqrt(x+h+5))/sqrt((x+h+5)(x+5)))$

$=lim_(h->0)1/(hsqrt((x+h+5)(x+5)))* (sqrt(x+5)-sqrt(x+h+5))*((sqrt(x+5)+sqrt(x+h+5))/(sqrt(x+5)+sqrt(x+h+5)))$

$=lim_(h->0)1/(hsqrt((x+h+5)(x+5)))* ((sqrt(x+5))^2-(sqrt(x+h+5))^2)/(sqrt(x+5)+sqrt(x+h+5))$

$=lim_(h->0)1/(hsqrt((x+h+5)(x+5)))* (x+5-x-h-5)/(sqrt(x+5)+sqrt(x+h+5))$

$=lim_(h->0)1/(hsqrt((x+h+5)(x+5)))* (-h)/(sqrt(x+5)+sqrt(x+h+5))$

$=lim_(h->0)1/sqrt((x+h+5)(x+5))* (-1)/(sqrt(x+5)+sqrt(x+h+5))$

$=1/sqrt((x+0+5)(x+5))* (-1)/(sqrt(x+5)+sqrt(x+0+5))$

$=1/sqrt((x+5)^2)* (-1)/(2sqrt(x+5))$

$=1/(x+5)* (-1)/(2sqrt(x+5))$

$=-1/(2(x+5)^(3/2))$