  We consider the following nonlinear Schr\"{o}dinger equation with derivative:
\begin{equation} iu_t =-u_{xx} -i |u|^{2}u_x -b|u|^4u , \quad (t,x) \in
\mathbb{R}\times\mathbb{R}, \ b \in\mathbb{R}. \end{equation} If $b=0$, this
equation is a gauge equivalent form of the well-known derivative nonlinear
Schr\"{o}dinger (DNLS) equation. The equation for $b\ge 0$ has degenerate
solitons whose momentum and energy are zero, and if $b=0$, they are algebraic
solitons. Inspired from the works on instability theory of the $L^2$-critical
generalized KdV equation, we study the instability of degenerate solitons in a
qualitative way, and when $b>0$, we obtain a large set of initial data yielding
the instability. The arguments except one step in our proof work for the case
$b=0$ in exactly the same way, and in particular the unstable directions of
algebraic solitons are detected. This is a step towards understanding the
dynamics around algebraic solitons of the DNLS equation.
