  We consider the approximation of manifold-valued functions by embedding the
manifold into a higher dimensional space, applying a vector-valued
approximation operator and projecting the resulting vector back to the
manifold. It is well known that the approximation error for manifold-valued
functions is close to the approximation error for vector-valued functions. This
is not true anymore if we consider the derivatives of such functions. In our
paper we give pre-asymptotic error bounds for the approximation of the
derivative of manifold-valued function. In particular, we provide explicit
constants that depend on the reach of the embedded manifold.
