  In a fibre bundle, natural derivatives of a section are defined as tangent
vector fields on the image of a section of the fibre bundle. A local extension
to vector fields in the tangent bundle leads to a direct proof of the formula
expressing the curvature of a connection in terms of covariant derivatives. The
result is based on a tensoriality argument and extends to nonlinear connections
on fibre bundles a well-known formula for linear connections on vector bundles.
