Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Exponential growth and decay

  1. Apr 28, 2010 #1

    for continuous compouding (in the exponential growth and decay) the function for interest after t years is:

    A(t) = A0*e^rt
    A0 should be read A at time 0

    in the book, they say when you differentiate this function this is what you get

    dA/dt = r*A0*e^rt = rA(t)

    I dont see how to do this differentiation to finally come up with rA(t). I am getting something else using implicit differentiation.


  2. jcsd
  3. Apr 28, 2010 #2
    [tex] A(t) \ = \ A_0 e^r^t [/tex]

    [tex] A'(t) \ = \ rA_0 e^r^t [/tex]

    A_0 is just a constant. [tex] A_0 e^r^t \ = \ A(t) [/tex] so in the second version they've just rewritten the derivative as r times the original function : [tex] A'(t) \ = \ rA_0 e^r^t \ = r A(t) [/tex] . There's no need to use implicit differentiation, I'm curious why you thought you needed to use it?
  4. Apr 28, 2010 #3
    Thanks for the reply.

    I now realize that impl diff doesnt make any sense (lets call it the severe flu I am fighting). The question I still have is this:

    d/dt A0*e^rt
    take A0 out and differentiate e^rt with respect to t
    Chain ruls has d/dt e^rt = e^rt * d/dt rt
    d/dt rt (using product rule) = r*d/dt t + t* d/dt r = r + t dr/dt

    Where does the 'r' from (as r times the original function) come from. What is wrong with my differentiation. I am not getting simply r * the original function as the derivative.


  5. Apr 28, 2010 #4
    Both methods work and in both methods you'll notice that r is a constant so it will drop out.

    I'll just tell you that you wouldn't need to use the product rule when you're differentiating a constant times a variable but it is handy to notice you can use both methods for the future.

    From reading this I don't think you understand the chain rule properly, not only should you notice that you're differentiating with respect to time (t) but that everything else in the function is just a constant.


    Some helpful videos if you're still a bit unsure just watch the ones relevant to you then come back and ask anything else if you're still a bit confused.

    Take Care :)
  6. Apr 29, 2010 #5
    That is exactl it! I didnt realize that r in this case was a constant. That totally makes sense now. The differentiation with respect to t of rt would simply be r.

    much appreciated! Thanks for the links as well.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook