Given a Riemannian manifold $M$, $V,W$ are smooth vector fields, $\phi_t$ is the flow of $X$. Show that $\frac{d}{dt}\langle D\phi_t(V), D\phi_t(W)\rangle = \langle L_X V, W\rangle +\langle V, L_XW\rangle+X\langle V, W\rangle$.
This does not make any sense to me:
$\frac{d}{dt}\langle D\phi_t(V), D\phi_t(W)\rangle = \langle \frac{D}{dt}D\phi_t(V）, W\rangle +\langle V, \frac{D}{dt}D\phi_t(W）\rangle$. Where can the third term possibly come from?
It comes from where you evaluate the inner product.
$\phi_t\colon M\to M$ is the flow of $X$, so $(d\phi_t)_p\colon T_pM\mapsto T_{\phi_t(p)}M$ and $$ \langle (d\phi_t)V, (d\phi_t)W\rangle\colon p\mapsto \langle (d\phi_t)_p V_p, (d\phi_t)_p W_p\rangle_{\phi_t(p)} $$ so when differentiating with respect to $t$ (at $t=0$), you also need to take account of the change in the point of evaluation $\phi_t(p)$.