q(t) = a / 1 + be^-kt
where b = a / q(initial) - 1 and k > 0 is constant.
Assuming a > q(initial) > 0, prove that the function q is everywhere increasing by analyzing its first derivative. Do not use specific values for any of the constants.
I found the derivative of q(t) to be
q'(t) = (-a(be^-kt)-bk) / ((1+be^-kt)^2)
First of all, is that right? And if it is, how do I prove that the function is everywhere increasing without using specific values for any of the constants?
Quotient Rule: (Derivative of Top(Bottom) - Top(Derivative of Bottom)) / ((Bottom)^2)
Derivative of Top: 0
Derivative of Bottom: 0 + be^-kt (I think this is where my problem is)
q'(t) = (0(1+be^-kt)-a(be^-kt)) / ((1+be^-kt)^2)
= (-a(be^-kt)) / ((1+be^-kt)^2)