$M$ a complete Riemannian manifold with nonpositive sectional curvature. Show that $|d(\exp_p)_v(w)| \geq |w|$ for all $v \in T_p M$, and all $w \in T_v(T_p M)$.
Update:
Based on the hints I've gotten, here is my attempted solution.
We will compare $M$ with $\tilde{M}=T_p M \approx T_v T_p M$ via the Rauch comparison theorem. Let $\gamma:[0,1] \rightarrow M$ be a geodesic and let $\tilde{\gamma}: [0,1] \rightarrow \tilde{M}$ be a comparison geodesic (which means that $\gamma$ and $\tilde{\gamma}$ have the same speed). Note that $\tilde{\gamma}$ also has no conjugate points because curvature zero. Write $\gamma(0)=p$ and $\gamma'(0)=v$.
Let $J(t)$ be the Jacobi field along $\gamma$ with $J(0)=0$ and $J'(0)=w$; then $J(t)$ can be written
$J(t)=d(exp_p)_{t\gamma'(0)}(tJ'(0))$
so we see that $J(1)=d(exp_p)_vw$.
The Rauch comparison theorem gives us that, for a Jacobi field $\tilde{J}$ along $\tilde{\gamma}$ with
$\tilde{J}(0)=J(0)=0$
$|\tilde{J}'(0)|=|J'(0)|$
$\langle \tilde{J}'(0),\gamma'(0) \rangle = \langle J'(0), \gamma'(0) \rangle$
we have $|\tilde{J}| \leq |J|$, so if we had a Jacobi field $\tilde{J}$ along $\tilde{\gamma}$ with $\tilde{J}(1)=w$ we'd be done.
There is a theorem that says if there is no conjugate points along $\tilde{\gamma}$ then there is a unique Jacobi field $\tilde{J}$ along $\tilde{\gamma}$ with $\tilde{J}(0)=0$ and $\tilde{J}(1)=w$ but I am unsure how how to get that the highlighted conditions (2) and (3) above hold.
