  We prove that the derivative map $d \colon \mathrm{Diff}_\partial(D^k) \to
\Omega^kSO_k$, defined by taking the derivative of a diffeomorphism, can induce
a nontrivial map on homotopy groups. Specifically, for $k = 11$ we prove that
the following homomorphism is non-zero: $$ d_* \colon
\pi_5\mathrm{Diff}_\partial(D^{11}) \to \pi_{5}\Omega^{11}SO_{11} \cong
\pi_{16}SO_{11} $$ As a consequence we give a counter-example to a conjecture
of Burghelea and Lashof and so give an example of a non-trivial vector bundle
$E$ over a sphere which is trivial as a topological $\mathbb{R}^k$-bundle (the
rank of $E$ is $k=11$ and the base sphere is $S^{17}$.)
  The proof relies on a recent result of Burklund and Senger which determines
those homotopy 17-spheres bounding $8$-connected manifolds, the plumbing
approach to the Gromoll filtration due to Antonelli, Burghelea and Kahn, and an
explicit construction of low-codimension embeddings of certain homotopy
spheres.

Geometric Topology;Algebraic Topology