Question

How do you find the derivative of: `f(x)=sqrt(x+1)`, using the limit definition?

Answer 1

The details depend on whether you use

`f'(a) = lim_(xrarra) (f(x) - f(a))/(x-a)`

or,

`f'(x) = lim_(hrarr0) ( f(x+h) - f(x))/(h)`.

For `f(x) = sqrt(x+1)`

You will need to evaluate one of:

`f'(a) = lim_(xrarra) (sqrt(x+1) - sqrt(a+1))/(x-a)`

or,

`f'(x) = lim_(hrarr0) ( sqrt(x+1+h) - sqrt(x+1))/(h)`.

Using either from of the definition, you'll use:

`(sqrtu-sqrtv)(sqrtu+sqrtv)=u-v`

For the first form of the definition, we get:

`f'(a) = lim_(xrarra) (sqrt(x+1) - sqrt(a+1))/(x-a)`

`=lim_(xrarra) ((sqrt(x+1) - sqrt(a+1)))/((x-a)) ((sqrt(x+1) +sqrt(a+1)))/((sqrt(x+1) + sqrt(a+1)))`

`= lim_(xrarra) ((x+1) - (a+1))/((x-a)(sqrt(x+1) + sqrt(a+1)))`

`= lim_(xrarra) 1/(sqrt(x+1) + sqrt(a+1)) = 1/(2sqrt(a+1)`

Using the second form of the definition is similar, but you'll end up with:

`f'(x) = lim_(hrarr0) ( sqrt(x+1+h) - sqrt(x+1))/(h)`

`= lim_(hrarr0) ( (x+1+h) - (x+1))/(h( sqrt(x+1+h) + sqrt(x+1)))`

`= lim_(hrarr0) (h)/(h( sqrt(x+1+h) + sqrt(x+1))) = 1/(2sqrt(x+1))`