The function $f:A \subseteq \mathbb{R}^n→\mathbb{R}^m$is differentiable at the point Definition : The function $f:A \subseteq  \mathbb{R}^n→\mathbb{R}^m$is differentiable at the point $a$ if there exists a linear transformation $T:\mathbb{R}^n→\mathbb{R}^m $ that satisfies the condition :
$$\lim\limits_{x \to a} \frac{ \parallel f(x)-f(a)-T(x-a)  \parallel }{ \parallel  x-a\parallel }=0$$
Or :
$$f(x)=f(a)+T(x-a)+r(x) \ \ \ \ \ : \lim\limits_{x \to a} \frac{\parallel r(x)  \parallel }{\parallel  x-a\parallel }=0$$
The $m×n$ matrix associated with the linear transformation $T$ is the matrix of partial derivatives, which we denote by $Df(a)$. We can refer to $Df(a)$ as the total derivative (or simply the derivative) of $f$.

Now :

Definition : The function $f:A \subseteq  \mathbb{R}→\mathbb{R}$ is differentiable at the point $a$ if there exists a linear transformation $T:\mathbb{R}→\mathbb{R} $ that satisfies the condition :
$$\lim\limits_{x \to a} \frac{ \parallel f(x)-f(a)-T(x-a)  \parallel }{ \parallel  x-a\parallel }=0$$
Or :
$$f(x)=f(a)+T(x-a)+r(x) \ \ \ \ \ : \lim\limits_{x \to a} \frac{\parallel r(x)  \parallel }{\parallel  x-a\parallel }=0$$

Is it right ?
but why in here :
Absolute value is not !!!!
Because, in $\mathbb R$, $f(x) \to 0$ if and only if $|f(x)| \to 0$.
If $f(x) = |x|$, try writing down which linear function $df_0$ would be.  Probably either $df_0(x) = x$ or $df_0 = -x$.  Let's say $df(x) = x$.  Then we evaluate the one sided limits:
\begin{eqnarray}
\lim_{x \to 0^-} \frac{f(x) - f(0) - df_0(x - 0)}{x - 0} &amp;=&amp;
\lim_{x \to 0^-} \frac{|x| - 0 - (x - 0)}{x - 0}\\
&amp;=&amp;\lim_{x \to 0^-} \frac{(-x)  - x }{x }\\
&amp;=&amp; -2
\end{eqnarray}
and
\begin{eqnarray}
\lim_{x \to 0^+} \frac{f(x) - f(0) - df_0(x - 0)}{x - 0} &amp;=&amp;
\lim_{x \to 0^+} \frac{|x| - 0 - (x - 0)}{x - 0}\\
&amp;=&amp;\lim_{x \to 0^+} \frac{x  - x }{x }\\
&amp;=&amp; 0.
\end{eqnarray}
So we see that the limit does not exist if $df_0(x) = x$.  You'll get a similar thing if you try $df_0(x) = -x$, or any other linear function.