Question

Using the limit definition, how do you find the derivative of `f(x) = sqrt(x + 2)`?

Answer 1

`f'(x)=1/(2sqrt(x+2))`

Explanation:

`f'(x)=lim_(hto0)(f(x+h)-f(x))/h`

`=lim_(hto0)(sqrt(x+h+2)-sqrt(x+2))/h`

`=lim_(hto0)((sqrt(x+h+2)-sqrt(x+2))(sqrt(x+h+2)+sqrt(x+2)))/(h(sqrt(x+h+2)+sqrt(x+2))`

`=lim_(hto0)(x+h+2-(x+2))/(h(sqrt(x+h+2)+sqrt(x+2))`

`=lim_(hto0)cancel(h)/(cancel(h)(sqrt(x+h+2)+sqrt(x+2)`

`=1/(sqrt(x+2)+sqrt(x+2))=1/(2sqrt(x+2))`