Solution of the logistic DE P'=kP(C-P)

[DMOC]

Homework Statement

Verify by direct cauculation that if k, C, and d are constants, then the function P(t) = C/(1+d*e$$^{-kCt}$$) is a solution of the logistic DE P' = kP(C-P).

Homework Equations

I don't think there are any for this problem. :)

The Attempt at a Solution

Okay, so ... uh ... I guess in this problem I should just be looking for the derivative of the original equation. So here goes ...

P(t) = C/(1+d*e$$^{-kCt}$$)
P(t) = C(1+d*e$$^{-kCt}$$)$$^{-1}$$ -- [I just moved the bottom part to the top.]
P(t) = -(e$$^{-t}$$)$$^{-2}$$*-1 (chain rule) <-- I think this is where I go wrong. C, k, and d are constants so I just made their derivaties one. Is that the right thing to do? Because somehow I get the feeling that the third line of work here isn't going to get me to the answer.

 

[lippyka]

P'(t) = -2e^-kCt(1+d*e^{-kCt})^{-3}Now I'm stuck. Could someone please tell me how to get from the fourth line to the original equation? Thank you so much!