# Derivation of developer productivity decline function

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) $\Large P(t) = P_0 \cdot e^{-rt}$