So I'm studying Essential Mathematics for Economic Analysis (Sydsæter, Hammond; page 180) now and stumbled upon this example aimed at illustrating why the Product rule for differentiation works the way it works. I have no problem understanding the formal proof which is given after that example, but the latter drives me crazy.
I would really appreciate if someone helped me with the questions I have. Here's a screenshot from the textbook.
Okay, so we have a formula for the revenue R(P)=P*D(P). Let's say P increases by one dollar and the R changes. That's comprehensible. Why does R increase? Well, R(P) increases by 1*D(P), 'cause each of the D(P) units brings in an extra dollar. But D(P) is also a function and a change in P by one dollar changes the D like: D(P+1)-D(P), which is close to D'(P) That's also more or less clear. But after that I am completely lost.
The (positive) loss due to a one dollar increase in the price per unit is then−P*D'(P), which must be subtracted from D(P) to obtain R'(P), as in equation.
- Why do we have have positive loss?
- What does it even mean?
- And why is this loss -P*D'(P)? Why negative?
- And why do we have to subtract it from D(P) to obtain R'(P)? I know that the derivative of R(P)=PD(P) is R'(P)=D(P)+PD'(P). But if I didn't know the derivative why would I subtract this "positive loss" from D(P) to obtain R'(P)?
I hope somebody would be kind enough to answer my ignorant questions. Thanks! :)