Derivation of developer productivity decline function

The first assumption is that developer productivity declines with a constant rate $r$:

(1) ${\frac {dP(t)}{dt}}$ = $-r\cdot P(t)$

Divide by $P(t)$:

(2) ${\frac {1}{P(t)}}$ $\cdot$ ${\frac {dP(t)}{dt}}$ = $-r$

(3) ${\frac {dP(t)}{P(t)}}$ = $-rdt$

(4) $\int {\frac {dP(t)}{P(t)}}$ = $-r\int tdt$

(5) $ln(P(t))=-rt+C$

Exponantiate both sides:

(6) $P(t)=e^{-rt}e^{C}$

(7) $P(t)=e^{C}\cdot e^{-rt}=e^{C}\cdot 1$

As P(0) = P_0 we get:

(8) $P(0)=e^{C}\cdot e^{-r\cdot 0}=e^{C}\cdot 1$

So $P(0)=e^{C}$ giving: (9) $P(t)=P_{0}\cdot e^{-rt}$

See: http://en.wikipedia.org/wiki/Exponential_decay#Solution_of_the_differential_equation