Derivative of a function with modulus terms. Please any one help me in calculation of $f(x)$, where $f(x)=-i\hbar\frac{\partial}{\partial x}\left(e^{-\frac{\left|x\right|}{a}}\right)$. Here $\hbar$ and $a$ is constant. Also scetch the $f(x)$.

$f(x)=-i\hbar\frac{\partial}{\partial x}\left(e^{-\frac{\left|x\right|}{a}}\right)$
It is sufficient to apply chain rule.
$\newcommand{\sign}{\text{sign }}$
$f(x)=-i\hbar e^{-\frac{\left|x\right|}{a}}\cdot\left(-\frac{1}{a}\sign x\right)=i\frac{\hbar}{a}(\sign x)e^{-\frac{\left|x\right|}{a}}\\\forall x\in\mathbb{R}\land x\ne0$
where $\;\sign x= \left \{ \begin{array}{rl}
1\;\;\;\;\;\text{ if }x>0\\
-1\;\;\;\;\;\text{ if }x<0
\end{array}
\right.$
$f(x)$ is not defined at $x=0$ and
$\lim_\limits{x\to0^-}f(x)=-i\frac{\hbar}{a}$ ,
$\lim_\limits{x\to0^+}f(x)=i\frac{\hbar}{a}$ .
The graph of the function $f(x)$ for $a=1.86$ is the following:

