Differentiation challenge/problem

[The Angelus]

Hello,

 

I am stuck on this problem right here:

 

Find the first and second derivative of: x(t) = KC/(C + e^-rt), where K, C and r are constants.

 

Any idea on how you would do this? Is there some trick that I need to learn to be able to do this, or am I just looking at it the wrong way.

 

 

Thanks in advance,

 

Atham

 

 

EDIT: Apparently the second deriv. simplifies down to -rt = ln C. I am not so sure though.

Edited April 9, 2015 by Atham

[Emilia1320]

I would solve that simply with differentation formulas.

First use quotient rule that derivative of f/g equals (f'g-g'f)/g^2 to obtain first derivative that is (rKCe^-rt)/(C+e^-rt)^2

Then use same rule to first derivative to obtain second derivative (this one is ugly) that is: (-r^2 KCe^-rt (C+e^-rt)^2+2(C+e^-rt)*r^2 Ke^-2rt C)/(C+e^-rt)^4

This form is unsimplified and propabky I've dropped a number somewhere, altough I don't see this simplifying to a linear form.

Remeber that derivative of e^f(x) is e^f(x)*f'(x)

[The Angelus]

Yeah I did that. When I have it equal zero to find the point of inflection, I get e^-rt = (c-1). I must have done something wrong, but it is quite close. 

 

EDIT: WAIT. I think I know what I did wrong.

[Emilia1320]

Yeah I did that. When I have it equal zero to find the point of inflection, I get e^-rt = (c-1). I must have done something wrong, but it is quite close. 

 

EDIT: WAIT. I think I know what I did wrong.

Wait, why you need point of infliction? If the question was only about first and second derivative ?

[The Angelus]

That was the next step in the book I was looking at. They got the second derivative to try and get the point of inflection when x''(t) = 0. I was trying to find out how they got their with the algebra and calculus.

 

BTW, how are you doing 3 sciences? I am so envious, since in our school you can only have 2 sciences and have to have one humanity.

[Emilia1320]

That was the next step in the book I was looking at. They got the second derivative to try and get the point of inflection when x''(t) = 0. I was trying to find out how they got their with the algebra and calculus.

 

BTW, how are you doing 3 sciences? I am so envious, since in our school you can only have 2 sciences and have to have one humanity.

Remember that to find a zero point of rational expression you just need to find a zero point for the nominator. Rational expression containing only real numbers has the value zero if and only if value of nominator is zero.

I have 3 sciences with special permission from IBO. It's possible to get with a reason, and it's common on Scandinavia since medicine here practically wants also physics.

[The Angelus]

Yeah, I just look at the numerator. 

 

I also wanted to study physics, since many medicine applications require physics. I should have gotten that too, now I am stuck with history (although not that bad). 

[Emilia1320]

Yeah, I just look at the numerator. 

 

I also wanted to study physics, since many medicine applications require physics. I should have gotten that too, now I am stuck with history (although not that bad).

Sorry to hear, and sorry if "looking at numerator" was too obivious, lol, I just occasionally forget that and get stupidly stuck so I wanted to make sure it's not that ^^

History is in my opinion terribly boring