# Variation of covariant derivative

In order to derive the equation of motion for a general action of the form

$$A = \int L \ dv$$

where the lagrangian function $$L = L ( \ T^{a ... b}_{\quad c...d} \ , \nabla_{\nu} T^{a ... b}_{\quad c...d})$$ depends only upon a tensor field and its covariant derivative.

I tried to derive the Euler Lagrange equations of motion but I'm stuck with the variation of the covariant derivative.

In S.Hawking-Ellis The Large Scale Structure of Spacetime, pag. 65, it is stated that $$\delta \ \nabla \ T = \nabla \ \delta \ T$$ for a given variation of the tensor fields $$T \longrightarrow T + \delta T$$

Now for simplicity consider the case of a single vector fields. First of all how do I define $$\delta T$$? And then how to prove that the variation commutes with the covariant differentiation?