Exponential Growth and Decay: Differentiating with Respect to Time

[ivan77]

Hello,

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 don't see how to do this differentiation to finally come up with rA(t). I am getting something else using implicit differentiation.

Thanks,

Ivan

 

[sponsoredwalk]

[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?

 

[ivan77]

[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?

Thanks for the reply.

I now realize that impl diff doesn't 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.

Thanks,

Ivan

 

[sponsoredwalk]

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.

www.khanacademy.org
http://www.ltcconline.net/greenl/courses/105/videos/VideoIndex.htm
http://online.math.uh.edu/HoustonACT/videocalculus/index.html

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 :)

 

[ivan77]

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.

www.khanacademy.org
http://www.ltcconline.net/greenl/courses/105/videos/VideoIndex.htm
http://online.math.uh.edu/HoustonACT/videocalculus/index.html

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 :)

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.

Ivan