Covariant derivative on principal bundle I know that there exists a connection on a principal bundle and via parallel transport it is possible to define a a covariant derivative on the associated bundle.
However, can we also define a covariant derivative on the principal bundle. I.e. something that can differentiate a section along a vector field? Or do we need a linear structure like the one in a vector bundle to 'take derivatives'?

It is defind in Wikipedia as $D\phi(v_0,...,v_k)=d\phi(hv_0,...,hv_k)$ where $h$ is the projection to horizontal subspace according to the given principal connection of the principal G-bundle $\pi:P\to M$.
If $\rho:G\to GL(V)$ is a representaion of $G$ on some vector space $V$, then a tensorial (or basic, i.e. G-equivariant and horizontal) k-form of type $\rho$ on $P$ can be identified with $P×_\rho V$-valued k-form on M.
As other answers/comments said, it is not a covariant derivative in the common sense, but is a covariant derivative in the sense that if $\phi$ is a $P×_\rho V$-valued k-form on M, and it is identified with tensorial form $\hat\phi$ on $P$, and $\nabla\phi$ is $P×_\rho V$-valued k+1-form on M, where $\nabla$ is the covariant derivative on $P×_\rho V$ corresonding to the given principal connection, and $\widehat{\nabla\phi}$ is the identification of $\nabla\phi$ to tensorial form on $P$, then $D\hat\phi=\widehat{\nabla\phi}$.
