My https://www.amazon.com/dp/0073532320 (p. 176 Example 7.1) pointed out that an investment ##p(t) = 100\,2^t## (##t## in year) that doubles the capital every year starting with an initial capital of $100, has an (instantaneous) rate-of-change ##\frac{\text{d}}{\text{d}t} p(t) = p'(t) = 100\,2^t\,\ln 2##.(adsbygoogle = window.adsbygoogle || []).push({});

Layman usually uses the formula ##p(t/12) = 100\,2^{t/12}## if he wants to know his capital at the end of each month. And, I think that makes sense because if the capital is never cashed, at the end of the first year, the initial capital doubles exactly to $200.

Now, the textbook claims that ##\frac{p'(t)}{p(t)} = \ln 2 \approx 69.3%## is the percentage change per year, and that should be surprising to most people because a percentage rate of 69.3% will double the investment each year if compounded "continuously".

Why it should make sense at all? I think people simply use something like ##p(t/12) = 100\,2^{t/12}## to calculate their capital after certain months. Why would knowing the instantaneous rate of change be beneficial at all in business, and so, should surprise people by the percentage change of 69.3% per year? Is the example simply a thought exercise without any direct connection to the real world?

Thank you.

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# [Compound Interest] Layman way vs. Derivative way

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