  In this paper, we explore the integrable fractional derivative nonlinear
Schr\"odinger (fDNLS) equation by using the inverse scattering transform.
Firstly, we start from the recursion operator and obtain a formal fDNLS
equation. Then the inverse scattering problem is formulated and solved through
the matrix Riemann-Hilbert problem. Subsequently, we give the explicit form of
the fDNLS equation according to the properties of squared eigenfunctions, such
as squared eigenfunctions are the eigenfunctions of the recursion operator of
the integrable equations. The reflectionless potential with a simple pole for
the zero boundary condition is carried out explicitly by means of determinants.
Finally, for the fractional one-soliton solution, we analyze the wave
propagation direction and the effect of the small fractional parameter
$\epsilon$ on the wave. The fractional one-soliton solution has been verified
rigorously. In addition, we also analyze the fractional rational solution
obtained by taking the limit of the fractional one-soliton solution.
