Let $f(x) = \frac{x}{1 + |x|}$ be differentiable at . . . .

Let $f(x) = |x|$. Then which of the following is true?

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be functions satisfying $f(x+y)=f(x)+f(y)+f(x)f(y)$ and $f(x)=x g(x)$ for all $x, y \in R$. If $\lim _{x \rightarrow 0} g(x)=1$,then which of the following statements is/are $TRUE$?
$(A)$ $f$ is differentiable at every $x \in R$
$(B)$ If $g(0)=1$,then $g$ is differentiable at every $x \in R$
$(C)$ The derivative $f^{\prime}(1)$ is equal to $1$
$(D)$ The derivative $f^{\prime}(0)$ is equal to $1$

The function $f(x) = (x - a)^2 \cos \frac{1}{(x-a)}$ for $x \neq a$ and $f(a) = 0$,is

The set of points where the function $f(x)=|x-1| e^{x}$ is differentiable,is

$f(x)= \begin{cases} 2a-x & \text{in } -a < x < a \\ 3x-2a & \text{in } a \leq x \end{cases}$
Then,which of the following is true?