  This letter is dedicated to providing proof of two statements concerning the
gradient expansion of relativistic hydrodynamics. The first statement is that
\textit{the ordering of transverse derivatives is irrelevant in the gradient
expansion of a non-conformal fluid}. The second statement is that \textit{the
longitudinal projection of the Weyl covariant derivative can be eliminated in
the gradient expansion of a conformal fluid}. This second statement does not
apply to curvature tensors. In the conformal case, we know that the ordering of
Weyl covariant derivatives is irrelevant in the gradient expansion.
