I am reading the proof of Bishop-Gromov's comparison theorem in Schoen and Yau's Differential Geometry book. They do it using the Jacobi fields approach. There is a step which I have trouble understanding.
The result that they use to get to the theorem is the following :
Theorem 1.2 If $M$ has a lower Ricci bound $Ric(M) \geq (n-1)K$, then $t^{n-1}A(t,\theta)S_K^{1-n}(t)$ is a non-increasing function of $t$.
Let me now describe what is this expression. You pick a point $x \in M$ and let $(t,\theta)$ be the polar coordinates in $T_xM$. The function $A(t,\theta)$ is the Jacobian of $exp_x$ at $(t,\theta) \in T_xM$, i.e. $\det (Dexp_x)_{(t,\theta)}$. One way to express $A(t,\theta)$ is to take $\gamma(t)$ a (unit speed) geodesic on $M$ starting at $x$ with $\dot{\gamma}(0) = v \in T_xM$, choose an orthonormal basis $\{v,w_2,\ldots, w_n\}$ of $T_xM$ and take $Y_i(t)$ to be the Jacobi field along $\gamma$ satisfying $Y_i(0) = 0, \dot{Y}_i(0)=w_i$. Then since $(Dexp_x)_{tv}(w_i) = t^{-1}Y_i(t)$ and $(Dexp_x)_{tv}(v) = \dot{\gamma}(t)$, the determinant of $(Dexp_x)_{tv}$ is $$t^{1-n}\det(\dot{\gamma}(t),Y_2(t),\ldots, Y_n(t)).$$
Now, on a manifold of constant sectional curvature $K$, if we do the same procedure, we can write down the the Jacobi fields explicitly: they are of the form $S_K(t)E_i(t)$ for $S_K(t)$ satisfying $\ddot{S}_K(t) = -KS_K(t)$ if $E_i(t)$ are the parallel extensions of the initial basis of $T_xM$. Hence the corresponding function $A_K(t,\theta)$ here is given explicitly by $t^{1-n}S_K^{n-1}(t)$.
Now, they say that Theorem 1.2 follows immediately from the Rauch Comparison Theorem. I don't see how that works. Rauch's theorem tells us that $||Y_i(t)|| \leq ||S_K(t)E_i(t)||$ for all $t$. So what we get from this is that $$ t^{n-1}A(t,\theta) = \det(\dot{\gamma}(t),Y_2(t),\ldots, Y_n(t)) \leq \prod ||Y_i(t)|| \leq S_K^{n-1}(t) $$ so that $$t^{n-1}A(t,\theta)S_K^{1-n}(t) \leq 1$$ for all $t$.
Am I close or do I need something else?