Mathematics in selection of RC time constant for an envelope detector for AM
What you need to understand here is not the exact words but the meaning which they are trying to convey. Here t0 is used to represent initial time(which is constant), ie starting time and t is anytime in future(which is a varibale). The same equation can be solved in many ways but here it's not the mathematics which is important but the idea he is trying to convey hence the approach.

For solving this you can take differentiation by substitution approach, ie put t-t0=p and then solve the equation and you will still get the same answer. The idea is don't get confused by t-t0, just treat them as a single variable.

PS:- This question is off topic, you can ask such question on mathematics platform.