When obtaining the Euler Lagrange equations for a scalar field with higher order derivatives in curved space is it the same to use $$ -\partial_\nu\partial_\mu\frac{\partial \sqrt{-g} \mathcal{L}}{\partial\partial_\nu\partial_\mu\phi}+\partial_\mu\frac{\partial \sqrt{-g} \mathcal{L}}{\partial\partial_\mu\phi}-\frac{\partial \sqrt{-g} \mathcal{L}}{\partial\phi}=0 $$ or $$ -\nabla\nu\nabla\mu\frac{\partial \mathcal{L}}{\partial\nabla\nu\nabla\mu\phi}+\nabla\mu\frac{\partial \mathcal{L}}{\partial\nabla\mu\phi}-\frac{\partial \mathcal{L}}{\partial\phi}=0~~? $$
For a canonical kinetic term it's easy to see that it's true.
