How can I prove the following equation: $\partial _q^{-1} f=\theta^{-1}(f)\partial_q^{-1}+\partial_q^{-1}\circ(\partial_q^*f)\circ \partial _q^{-1}$.

Where $\partial_q^{-1}$ is the formal inverse of the $q$-differential operator $\partial_q f(x)=\frac{f(qx)-f(x)}{(q-1)x}$, i.e. a integral operator, and $\theta$ is the $q$-shift operator,$\theta(f(x))=f(qx)$.