  Quaternion derivatives in the mathematical literature are typically defined
only for analytic (regular) functions. However, in engineering problems,
functions of interest are often real-valued and thus not analytic, such as the
standard cost function. The HR calculus is a convenient way to calculate formal
derivatives of both analytic and non-analytic functions of quaternion
variables, however, both the HR and other functional calculus in quaternion
analysis have encountered an essential technical obstacle, that is, the
traditional product rule is invalid due to the non- commutativity of the
quaternion algebra. To address this issue, a generalized form of the HR
derivative is proposed based on a general orthogonal system. The so introduced
generalization, called the generalized HR (GHR) calculus, encompasses not just
the left- and right-hand versions of quaternion derivative, but also enables
solutions to some long standing problems, such as the novel product rule, the
chain rule, the mean-valued theorem and Taylor's theorem. At the core of the
proposed approach is the quaternion rotation, which can naturally be applied to
other functional calculi in non-commutative settings. Examples on using the GHR
calculus in adaptive signal processing support the analysis.
