Let $X$ be a vector field and $F^t$ the corresponding locally defined flow on a smooth manifold $M$. Thus $F^t(p)$ is defined for small $t$ and the curve $t \mapsto F^t(p)$ is the integral curve for $X$ that goes through $p$ at $t=0$.
If $f : M \to \mathbb R$ is a function, then the Lie derivative $L_X f = D_X f = df(X)$ is given by $(L_X f)(p) = \lim_{t \to 0} \frac{f(F^t(p))-f(p)}t$.
For the analogue with vector fields, we have $(L_X Y)\vert_p = \lim_{t \to 0} \frac{DF^{-t}(Y\vert_{F^t(p)})-Y\vert_p}t$, where $t \mapsto DF^{-t}(Y\vert_{F^t(p)})$ is a curve that lies in $T_p M$.
I am studying the proof that $L_X Y = [X,Y]$ (the Lie bracket), where $X,Y$ are vector fields in $M$. One of the steps I do not understand is how the author (Petersen, Riemannian Geometry) obtained $$ D_{Y\vert_{F^t}} f - D_{DF^t(Y)} f = (D_Y f) \circ F^t - D_Y(f \circ F^t). $$ In particular, how a composition of functions appears from taking directional derivatives. Any clarification on this alone helps.
Where this equality in question occurs: Manifold Theory Notes by Petersen, page 43, lines (-5)-(-6). This same equality also appears on page 54, lines (-1)-(-2) of Riemannian Geometry, Third edition, by Peter Petersen.
I will give a few definitions I didn't see in those notes so had to guess:
$D_{Y\vert_{F^t}}$ is defined by: $$(D_{Y\vert_{F^t}}f)(p) = D_{Y\vert_{F^t(p)}}f$$
If $\phi:M\to N$, and $Y$ is a vector field $M\to TM$, then $D\phi(Y)$ is the vector field on $N$ given by $$D_{D\phi(Y)}(f)=D_Y(f\circ \phi).$$
$((D_Y f) \circ F^t)(p)$ is $(D_Y f)(F^t(p))$ which by definition of $D_Y$ is just $D_{Y\vert_{F^t(p)}}f$, which is the same as in the definition 1 above. So: $$D_{Y\vert_{F^t}}f=(D_Y f) \circ F^t.$$
And by definition 2 above with $\phi=F^t$, we have $D_{DF^t(Y)} f=D_Y(f\circ F^t)$.
Subtracting gives the equality you want ($D_{Y\vert_{F^t}}f-D_{D^t(Y)} f=(D_Y f) \circ F^t-D_Y(f\circ F^t).$)