the particles number through energy boundary in dt is dN
(E+dE)-------->|(E)-------->(E-dE)
boundary
the number of particles that pass through the energy boundary per unit of time,
J_E = \frac{dN}{dt} = \frac{dN}{dE}\frac{dE}{dt}
----------------->[(E+dE)------------------](E)--------------->
the change of particle number due to energy variation within energy interval [E,E+dE] per unit time if dE/dt<0.
\begin{equation} \begin{split} \frac{dN}{dt} & = dN/dE(E+dE)d(E+dE)/dt - dN/dE(E)dE/dt \\ & = [dN/dE(E)+\frac{d^2N}{dE^2}(E) dE] \cdot [dE/dt + \frac{d^2E}{dEdt}(E)dE ] - dN/dE(E)dE/dt \\ & = \frac{d}{dE}(\frac{dN}{dE}(E)\frac{dE}{dt}) \cdot dE \end{split} \end{equation}
thus, \frac{dN}{dEdt} = \frac{d}{dE}(\frac{dN}{dE}(E)\frac{dE}{dt}) = \frac{dJ_E}{dE}
the change of particle number due to motion