  We propose a nonparametric method to explicitly model and represent the
derivatives of smooth underlying trajectories for longitudinal data. This
representation is based on a direct Karhunen--Lo\`eve expansion of the
unobserved derivatives and leads to the notion of derivative principal
component analysis, which complements functional principal component analysis,
one of the most popular tools of functional data analysis. The proposed
derivative principal component scores can be obtained for irregularly spaced
and sparsely observed longitudinal data, as typically encountered in biomedical
studies, as well as for functional data which are densely measured. Novel
consistency results and asymptotic convergence rates for the proposed estimates
of the derivative principal component scores and other components of the model
are derived under a unified scheme for sparse or dense observations and mild
conditions. We compare the proposed representations for derivatives with
alternative approaches in simulation settings and also in a wallaby growth
curve application. It emerges that representations using the proposed
derivative principal component analysis recover the underlying derivatives more
accurately compared to principal component analysis-based approaches especially
in settings where the functional data are represented with only a very small
number of components or are densely sampled. In a second wheat spectra
classification example, derivative principal component scores were found to be
more predictive for the protein content of wheat than the conventional
functional principal component scores.
