$\nabla_Y(df)$ is either a (1,1)-tensor or (0,1)-tensor (which one is it by the way?), and so is $d(\nabla_Yg)$. Are these two actually the same thing if $f=g$?
$f$ and $g$ are both some functions on a smooth manifold, $\nabla_Y$ is the affine connection (some vector field Y) and d is the exterior derivative.