I am trying to interpret what the attached differential equation for t_p_x means. In this case (when differentiating with respect to t), we are interested in how the probability of survival (from age x to age x+t) changes with respect to time t. This will always be negative because we ultimately want the life to stay alive over time t, hence we need to multiply by the force of mortality of staying alive at time x+t (which is the negative mu term). In the notes, it states that the probability of surviving for age x to age x+t is a decreasing function of t. And that is why we have the minus sign at the front. What does this mean exactly? Thank you
"tpx is a decreasing function of t" means its graph would be downward-sloping with respect to t. Equivalently, it means the derivative of tpx with respect to t is negative. tpx is a decreasing function of time, since for a fixed aged x, the probability of surviving for t years decreases, as t increases.
