I work with the Annuity present value factor, which I want to differentiate with respect to r:
$\sum_{t=T_1}^{T_2} (1+r)^{-t}$ which equals $\frac{(1 + r)^{1 - T_1} - (1 + r)^{-T_2}}{r}$
If I use the derivation of the Sum-expression I can easily proof, that the derivative with respect to r ist <0. If i do the same with the other (but equal) term, after some transforming I am stuck with $(1 + r)^{1 + T_2} \cdot (1 + r \cdot T_1) < (1 + r)^{T_1} \cdot (1 + r \cdot(1+ T_2))$ at which point I don't see how to proceed. How to solve this inequation? There must be some math rules or relations I am missing.
EDIT: The conditions are $0
EDIT: I tried to follow @Ross's instructions. I then end up with $\frac{(1+r\cdot(1+T_2))}{(1+r\cdot T_1)} < (1+r)^{1+T_2-T_1}$ Unfortunately I still don't know how to proceed from there on.