  The main goal of the paper is to prove that the sequence of functions $f(x),
Df(x), \dots, D^{2n+1}f(x)$, where $f(x)$ is $x^n\sin x$ or $x^n\cos x$ are
linearly independent. Or more generally: that the sequence of functions
$D^kf(x), D^{k+1}f(x), \dots, D^{2n+k+1}f(x)$, $k\in \mathbb{N}$ is linearly
independent. The problem is solved by a suitable transformation of the matrix
of determinant of the Wronskian. Another approach for a special sequence of
derivatives of functions uses only the definition of linear independence of
functions. This approach generates interesting, non-elementary combinatorial
identities.
