Difference between revisions of "Derivation of developer productivity decline function"

Latest revision as of 19:51, 27 August 2020

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) $P(t)=P_{0}\cdot e^{-rt}$

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-Derivation of developer productivity decline function
+The first assumption is that developer productivity declines with a constant rate <math>r</math>:
+(1) <math>\frac{dP(t)}{dt}</math> = <math>-r \cdot P(t)</math>
+Divide by <math>P(t)</math>:
+(2) <math>\frac{1}{P(t)}</math> <math>\cdot</math> <math>\frac{dP(t)}{dt}</math> = <math>-r</math>
+(3) <math>\frac{dP(t)}{P(t)}</math> = <math>-r dt</math>
+(4) <math>\int \frac{dP(t)}{P(t)}</math> = <math>-r \int t dt</math>
+(5) <math>ln(P(t))= -rt +C </math>
+Exponantiate both sides:
+(6) <math>P(t) = e^{-rt}e^C </math>
+(7) <math>P(t) = e^C \cdot e^{-rt} =e^C \cdot 1</math>
+As P(0) = P_0 we get:
+(8) <math>P(0) = e^C \cdot e^{-r \cdot 0} = e^C \cdot 1</math>
+So <math>P(0) = e^C</math> giving:
+(9) <math>P(t) = P_0 \cdot e^{-rt}</math>
+See: http://en.wikipedia.org/wiki/Exponential_decay#Solution_of_the_differential_equation
 [[Category:math]]