Question

Question #75a4b

Answer 1

See below

Explanation:

The first principle defines the derivative of a function `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))`