Question

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

Answer 1

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

Explanation:

Using the limit definition of a derivative, we have

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

`= lim_(h->0)(2sqrt(x+h)-2sqrt(x))/h`

`= 2lim_(h->0)(sqrt(x+h)-sqrt(x))/h*(sqrt(x+h)+sqrt(x))/(sqrt(x+h)+sqrt(x))`

`=2lim_(h->0)(x+h-x)/(h(sqrt(x+h)+sqrt(x))`

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

`= 2*1/(sqrt(x+0)+sqrt(x))`

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

`=1/sqrt(x)`