On the differentiation of a composite function with a generalized vector argument on homogeneous time scales; pp. 309–322Full article in PDF format
The paper proves a theorem on the differentiation of a composite function with a generalized vector argument. The theorem is formulated in terms of the delta derivative, which in the case of homogeneous time scales incorporates both the ordinary derivative and the difference operator. The term “generalized vector argument” implies that a composite function is allowed to depend not only on some variables but also on their delta derivatives. A formula in the theorem shows how the higher-order delta and partial derivatives of a composite function commute. Moreover, it enables reducing the order of the delta derivative, making computations simpler and more efficient. The computational efficiency of the formula was analysed on the basis of experiments in the symbolic computation software Mathematica.
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