The first assumption is that developer productivity declines with a constant rate $r$:
(1) $\Large \frac{dP(t)}{dt}$ = $\large -r \cdot P(t)$
Divide by $P(t)$:
(2) $\Large \frac{1}{P(t)}$ $\cdot$ $\Large \frac{dP(t)}{dt}$ = $-r$
(3) $\Large \frac{dP(t)}{P(t)}$ = $-r dt$
(4) $\Large \int \frac{dP(t)}{P(t)}$ = $-r \int t dt$
(5) $\large ln(P(t))= -rt +C $
Exponentiate both sides:
(6) $\large P(t) = e^{-rt}e^C $
(7) $\large P(t) = e^C \cdot e^{-rt}$
As P(0) = P_0 we get:
(8) $\large P(0) = e^C \cdot e^{-r \cdot 0} = e^C \cdot 1$
So $\large P(0) = e^C$ giving:
(9) $\Large P(t) = P_0 \cdot e^{-rt}$

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