How to determine a function is increasing in time using derivative?

Question

I have this function p(t) = N / (99e^(-2*t)+1) where N = 2000000 that I need to derive to show it's increasing in time. I tried deriving it p'(t) = (396000000 e^(2 t))/(99+e^(2 t))^2 but when I plot the equation at some point it starts decreasing over time. What am I missing?

Thanks!

Answer 1

Show that the derivative function is positive for all values t>=0 then you have proven that the function is increasing.

What you can say about your derivative is that with any t>0, observe that it leads to a positive number being divided by a positive number, which implies the derivative is positive.

Answer 2

Hints:

• the value of $N$ is irrelevant (other than it being positive), since $\,p(t)\,$ and $\,\dfrac{1}{N} p(t)\,$ are directly proportional for any $\,N \gt 0\,$, and therefore have the same monotonicity;

• there is no need to use derivatives here, it's enough to notice that when $\,t\,$ increases, $\,e^{-2t}\,$ decreases, so $\,99e^{-2t}+1\,$ decreases, so $\displaystyle\, p(t) = \frac{1}{99e^{-2t}+1}\,$ increases.

but when I plot the equation at some point it starts decreasing over time. What am I missing?

Then something is wrong with your plot. What exactly is wrong can't be answered unless you provide some details on how you plot it and what you are getting.