I am trying to confirm a stated result on my lecture slide.
Question:
Given that $A:= \sum_i^n \frac{a_i}{(1+b)^{t_i}}$, where $a_i,b \in \mathbb{R}_+$ and $t_i \in \{t_1,...,t_n\}$ where $0 < t_1 < ... < t_n < \infty$.
Demonstrate that:
$-\frac1A \frac{\partial A}{\partial b} = - \sum_i^n t_i \frac{a_i}{(1+b)^{-t_i}}$
Current progress:
$\frac{\partial A}{\partial b} = - \sum_i^n t_i \frac{a_i}{(1+b)^{t_i + 1}}$
PROBLEM
Just from this first step (which could be incorrect), it seems that I can't arrive at what the lecture slide claims.
Note that this is not from a mathematics lecturer so it could be wrong.
Any assistance welcome and appreciated :-)