Dif­fer­en­ti­able phys­ic­ally-based ren­der­ing has be­come an in­dis­pens­able tool for solv­ing in­verse prob­lems in­volving light. Most ap­plic­a­tions in this area jointly op­tim­ize a large set of scene para­met­ers to min­im­ize an ob­ject­ive func­tion, in which case re­verse-mode dif­fer­en­ti­ation is the meth­od of choice for ob­tain­ing para­met­er gradi­ents. However, ex­ist­ing tech­niques that per­form the ne­ces­sary dif­fer­en­ti­ation step suf­fer from either stat­ist­ic­al bi­as or a pro­hib­it­ive cost in terms of memory and com­pu­ta­tion time. For ex­ample, stand­ard tech­niques for auto­mat­ic dif­fer­en­ti­ation based on pro­gram trans­form­a­tion or Wengert tapes lead to im­prac­tic­ably large memory us­age when ap­plied to phys­ic­ally-based ren­der­ing al­gorithms. A re­cently pro­posed ad­joint meth­od by Ni­mi­er-Dav­id et al. 2020 re­duces this to a con­stant memory foot­print, but the com­pu­ta­tion time for un­biased gradi­ent es­tim­ates then be­comes quad­rat­ic in the num­ber of scat­ter­ing events along a light path. This is prob­lem­at­ic when the scene con­tains highly scat­ter­ing ma­ter­i­als like par­ti­cip­at­ing me­dia. In this pa­per, we pro­pose a new un­biased back­propaga­tion al­gorithm for ren­der­ing that only re­quires con­stant memory, and whose com­pu­ta­tion time is lin­ear in the num­ber of scat­ter­ing events (i.e., just like path tra­cing). Our ap­proach builds on the in­vert­ib­il­ity of the loc­al Jac­obi­an at scat­ter­ing in­ter­ac­tions to re­cov­er the vari­ous quant­it­ies needed for re­verse-mode dif­fer­en­ti­ation. Our meth­od also ex­tends to spec­u­lar ma­ter­i­als such as smooth dielec­trics and con­duct­ors that can­not be handled by pri­or work.