Expert: Steve Holleran - 8/15/2008

Question
Hi Steve, you've helped me once before and I think I'm stuck again. Let me know if you can help.

dP/dt = kP(1-(P/L))

a) Use the chain rule to find d^2P/dt^2
b) Show that the point of diminishing returns, where d^2P/dt^2 = 0, occurs where P = L/2.

The answer in the back of the book for part "a" said:
k(1-(2P/L)) (dP/dt). I don't understand how they got there, or why dP/dt is a part of the answer. Seconldy, I don't understand how dP/dt = kP(1-(P/L)) can have P as the variable if it is the derivative of P with respect to t. Shouldn't t be the variable?

Answer
Hi Tim,

This is the way I see it:

   dP/dt = kP(1 - P/L) = kP - (k/L)P^2

Now you want to differentiate with respect to t, so whenever you come to a variable that is not t , (like P), you have to tack on a "dP/dt" :

  d^2P/dt^2 = k * dP/dt - (k/L) * 2P * dP/dt

            = k * (dP/dt - (2P/L)dP/dt)

            = k * (1 - (2P/L)) * dP/dt


Then for the second part, set this = 0

              k * (1 - (2P/L)) * dP/dt = 0

Now the only factor that could be 0 is (1 - (2P/L)), and then

           1 - (2P/L) = 0 ---> 1 = 2P/L--> L = 2P --> P = L/2


I think this is what is going on.
Steve