 minuziavj

2022-09-30

Units for rate of change, instantaneous or otherwise
If we multiply a principal amount A0 in dollars ($) by an interest rate r in percentage (%) once, let's say for a month (m), then we have an amount A in dollars per month ($/m)
$A\left(\text{/m}\right)={A}_{0}\left(\text{}\right)\cdot r\left(\text{%}\right)$
The problem is that for the devices to agree arithmetically, the hobby rate might ought to be in units of in keeping with month (/m). that is additionally the case if we multiply the hobby rate by using a time frame t in months per year (m/y), then we've an amount in dollars consistent with yr($/y) $A\left(\text{/y}\right)={A}_{0}\left(\text{}\right)\cdot r\left(\text{%}\right)\cdot t\left(\text{m/y}\right)$ Once again, the interest rate would have to be in units of per month (/m) which looks strange. This is still the case for the instantaneous rate of change of the amount with respect to time $\frac{dA}{dt}$ in dollars per period of time ($/m), which looks a lot like the formula for exponential growth and decay
$\frac{dA}{dt}\left(\text{/m}\right)={A}_{0}\left(\text{}\right)\cdot r\left(\text{%}\right)$
Another time, the hobby fee could need to be in units of per month(/m). i have continually considered a charge like a scaling element with out wondering plenty of the units, so how am i able to make feel of the gadgets for the above interest charge? Kaitlyn Levine

Your dimensions are correct, the proper interpretation to your interest rate is that it's the amount of money that you earn per month, scaled by the principal amount, so it simply has units of "per month". If you like, you could view it as a frequency (related to the frequency of doubling your investment etc.)
In general, a derivative $\frac{d}{dx}$ acts as if it has dimensions of $\frac{1}{x}$ - e.g. a velocity has units m/s and is given by $v=\frac{dx}{dt}$ where x has units of m and t has units of s. More complicated derivatives (such as partial derivatives, divergence, etc) also follow this pattern.

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