Question

Using the limit definition, how do you differentiate `f(x)=sqrt(x+1)`?

Answer 1

`lim_{Delta->0}((sqrt(x+1+Delta)-sqrt(x+1)))/Delta=1/(2sqrt(x+1))`

Explanation:

Let us calculate the limit indirectly.
First we calculate the limit:

`l(x)=lim_{Delta->0}(sqrt(x+1+Delta)+sqrt(x+1))times lim_{Delta->0}((sqrt(x+1+Delta)-sqrt(x+1)))/Delta`
This limit is equivalent to
`lim_{Delta->0}((x+1+Delta)-(x+1))/Delta=1`
but
`lim_{Delta->0}(sqrt(x+1+Delta)+sqrt(x+1)) = 2sqrt(x+1)`
Putting all together
`l(x)=2sqrt(x+1) lim_{Delta->0}((sqrt(x+1+Delta)-sqrt(x+1)))/Delta = 1`
and finally
`lim_{Delta->0}((sqrt(x+1+Delta)-sqrt(x+1)))/Delta=1/(2sqrt(x+1))`

Answer 2

See explanantion

Explanation:

To solve this you use 2 general principles.

`color(blue)("Principle 1:")`

Multiply any number by 1 and you do not change its value. However you can change the way it looks. Consider a basic case of `3xx1=3`

Multiply 3 by 1 but in the form of `1=sqrt(2)/sqrt(2) -> 3xxsqrt(2)/sqrt(2) =(3sqrt(2))/sqrt(2)`

'.......................................................................
`color(blue)("Principle 2:")`

Suppose you had `(a+b)`

Multiply it by 1 but in the form iof `1=(a-b)/(a-b)`

Then `(a+b)(a-b)/(a-b) = (a^2-b^2)/(a-b)`
`color(red)("Your question has square roots and to get rid of this you square it!")`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Given:`" "f(x)=sqrt(x+1)`

Write as `y=sqrt(x+1)` ..................................(1)

Increment your values giving

`y+delta y =sqrt(x+delta x+1)`.........................(2)

Equation (2) - Equation (1)

`delta y=sqrt(x+delta x + 1) - sqrt(x+1)`

Divide throughout by `delta x`

`(delta y)/(delta x) = (sqrt(x+delta x + 1) - sqrt(x+1))/(delta x)` .......(3)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Multiply equation (3) by 1 but where `1=(sqrt(x+delta x + 1) + sqrt(x+1))/(sqrt(x+delta x + 1) + sqrt(x+1))`

Now we have our `(a-b)(a+b)=a^2-b^2` condition giving:

`(delta y)/(delta x) = ((x+delta x + 1) - (x+1))/(deltax(sqrt(x+delta x + 1) + sqrt(x+1)))`

`(delta y)/(delta x) =( cancel(x)+cancel(delta x) +cancel( 1) -cancel( x)-cancel(1))/(cancel(deltax)(sqrt(x+delta x + 1) + sqrt(x+1)))`

`lim_(delta x->0) (deltay)/(deltax)= 1/(sqrt(x+1)+sqrt(x+1)) = 1/(2sqrt(x+1))`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Which is of form `d/(dx) (x^n) = nx^(n-1)`