Beaver

First off, all which follows is solely from what could be gleaned from online aerial recon of this stretch, along with topo analysis. As a result, take all info with appropriate caution. The following is not likely to be entirely accurate or complete.

Increased gradient actually starts about a half-mile upstream of the bridge. (Unknown how convenient it might be to get up there, nor weather there is really much reason to. It appears there may be gradient of about 65FPM in this half-mile, which may include a somewhat significant drop early on, but online-aerial view is greatly obstructed by trees -- perhaps the river is as well.)

At the put-in drainage area is about 15 square miles (to the best I can do using online planimeter and topomaps).

There is somewhere near 80' of drop in the first mile below the bridge. Again, online/aerial view is mostly very obscured, so no telling what may be in here -- could be good, could be nasty portage fest.

After some twists and turns, the river widens as Thirtynine Creek comes in from the left (North). This adds appreciable flow, and from here down, the Beaver is generally a much broader stream. The next (roughly) mile-and-a-half will be the lowest gradient of the run (~20FPM).

The flattest mile (just past) is precursor to the steepest mile of this section, which drops somewhere very near 100 FPM.

If it is possible to get to this point (unpaved double-track appears on aerial views -- unknown if it is private/gated or open), you could shorten the trip to just over five miles (knocking off two miles of much lower gradient, ~30FPM).