Based on the fractional $q$-integral with the parametric lower limit of
integration, we define fractional $q$-derivative of Riemann-Liouville and
Caputo type. The properties are studied separately as well as relations between
them. Also, we discuss properties of compositions of these operators.
Based on the fractional $q$-integral with the parametric lower limit of
integration, we define fractional $q$-derivative of Riemann-Liouville and
Caputo type. The properties are studied separately as well as relations between
them. Also, we discuss properties of compositions of these operators.