Question

Question #99bbe

Answer 1

`d/(dx) sqrt(x+4) = 1/( 2sqrt(x+4))`

Explanation:

Using the definition of derivative:

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

we have:

`d/(dx) sqrt(x+4) = lim_(h->0) (sqrt(x+4+h)-sqrt(x+4))/h`

Rationalize the numerator using the identity: `(a+b)(a-b) = (a^2-b^2)`

`d/(dx) sqrt(x+4) = lim_(h->0) ((sqrt(x+4+h)-sqrt(x+4))/h )((sqrt(x+4+h)+sqrt(x+4))/ (sqrt(x+4+h)+sqrt(x+4)))`

`d/(dx) sqrt(x+4) = lim_(h->0) ((x+4+h)-(x+4))/(h (sqrt(x+4+h)+sqrt(x+4)))`

`d/(dx) sqrt(x+4) = lim_(h->0) h/(h (sqrt(x+4+h)+sqrt(x+4)))`

`d/(dx) sqrt(x+4) = lim_(h->0) 1/( (sqrt(x+4+h)+sqrt(x+4))) = 1/( 2sqrt(x+4))`