A lot of partition of unity arguments have some map of the form $f=\sum_i \psi_if_i$. Is there a formula for the differential $df_p$ in terms of its summands?
For instance, suppose $f_i:U_i\to V_i$ are a family of maps where the $U_i$ cover a smooth manifold $M$, and the $V_i$ are subsets of $\mathbb{R}$. Let $\{\psi_i\}$ be a partition of unity subordinate the $\{U_i\}$, and it makes sense to define $f=\sum_i \psi_if_i$.
Can you write $df_p$ in terms of $d(f_i)_p$ and $d(\psi_i)_p$? Since it usually does not make sense to multiply maps on manifolds, I've never seen a rule for the differential of a product of maps.