Covariant derivative commutes with partial derivative?

Question

I'm trying to understand the proof of Proposition 2.4 in Chapter 9 of do Carmo's Riemannian Geometry. There's a part that I'm unsure of where, "by the symmetry of the Riemannian connection", he uses $$\frac{D}{dt}\frac{df}{ds}=\frac{D}{ds}\frac{df}{dt},$$ where $f:(-\varepsilon,\varepsilon)\times [0,1]\to M$ is a variation of the curve $c:[0,1]\to M$. The symmetry of the connection means $$\nabla_XY-\nabla_YX=[X,Y],$$ so I tried to justify his use of the first formula by first writing $$\frac{D}{dt}\frac{df}{ds}=\nabla_{\partial f/\partial t}\left(\frac{df}{ds}\right),$$ then saying $$\left[\frac{\partial f}{\partial t},\frac{\partial f}{\partial s}\right]=0,$$ and so $$\frac{D}{dt}\frac{df}{ds}=\nabla_{\partial f/\partial t}\left(\frac{df}{ds}\right)=\nabla_{\partial f/\partial s}\left(\frac{df}{dt}\right)=\frac{D}{ds}\frac{df}{dt}.$$ I know this is a simple thing; I just want to make sure that it's correct. Thanks.