  A general formulation of noncommutative or quantum derivatives for operators
in a Banach space is given on the basis of the Leibniz rule, irrespective of
their explicit representations such as the G\^ateaux derivative or commutators.
This yields a unified formulation of quantum analysis, namely the invariance of
quantum derivatives, which are expressed by multiple integrals of ordinary
higher derivatives with hyperoperator variables. Multivariate quantum analysis
is also formulated in the present unified scheme by introducing a partial inner
derivation and a rearrangement formula. Operator Taylor expansion formulas are
also given by introducing the two hyperoperators $ \delta_{A \to B} \equiv
-\delta_A^{-1} \delta_B$ and $d_{A \to B} \equiv \delta_{(-\delta_A^{-1}B) ;
A}$ with the inner derivation $\delta_A : Q \mapsto [A,Q] \equiv AQ-QA$.
Physically the present noncommutative derivatives express quantum fluctuations
and responses.
