1. ## domain of derivative

Domain of f(x)=sqrt(1+2x)
1+2x>=0
x>=-1/2
domain:: (-1/2,+inf)

and
domain of f '(x)=1/sqrt(1+2x).........do not understand to do this one..

2. Hello, Gracy!

A minor (but important) change in your first answer . . .

Domain of $f(x) = \sqrt{1+2x}$

$1 + 2x \:\geq \:0\quad\Rightarrow\quad x\:\geq \:-\frac{1}{2}$

Domain: . $\left[-\frac{1}{2},\:\infty\right)$
. . . . . . .

We include the $-\frac{1}{2}$

Domain of $f'(x) \:=\:\frac{1}{\sqrt{1+2x}}$

That prime-mark is distracting, but we can ignore it.

Since $1 + 2x$ is under a square root, it must be positive or 0.
But since it is in the denominator, it cannot equal 0.

Hence, we have: . $1 + 2x\:>\:0\quad\Rightarrow\quad x \:>\:-\frac{1}{2}$

Domain: . $\left(-\frac{1}{2},\:\infty\right)$

3. Basically whenever you have a continous function on [a,b] and differenciable at (a,b) then the derivative does not exist at 'a' and 'b'. Do you see why? One of the conditions that implies a function is not differenciable is when it fails to be countinous. Note whenever you have a continous function on [a,b] then it is continous from the right at 'a' and continous from the left at 'b' but it is not fully continous.