Units for rate of change, instantaneous or otherwiseIf 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 ( $/m ) = A 0 ( $ ) ⋅ r ( % )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 ( $/y ) = A 0 ( $ ) ⋅ r ( % ) ⋅ t ( m/y )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 d A d t in dollars per period of time ($/m), which looks a lot like the formula for exponential growth and decay d A d t ( $/m ) = A 0 ( $ ) ⋅ r ( % )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?

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Units for rate of change, instantaneous or otherwiseIf we multiply a principal amount A0 in dollars ($) by an interest rate r in percentage (%) once, let&#039;s say for a month (m), then we have an amount A in dollars per month ($/m)  A  (      $/m    )  =      A    0    (      $    )  ⋅  r  (      %    )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&#039;ve an amount in dollars consistent with yr($/y)  A  (      $/y    )  =      A    0    (      $    )  ⋅  r  (      %    )  ⋅  t  (      m/y    )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             d      A              d      t       in dollars per period of time ($/m), which looks a lot like the formula for exponential growth and decay            d      A              d      t        (      $/m    )  =      A    0    (      $    )  ⋅  r  (      %    )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 · Accepted Answer

Your dimensions are correct, the proper interpretation to your interest rate is that it&#039;s the amount of money that you earn per month, scaled by the principal amount, so it simply has units of &quot;per month&quot;. If you like, you could view it as a frequency (related to the frequency of doubling your investment etc.)In general, a derivative       d          d      x       acts as if it has dimensions of       1    x   - e.g. a velocity has units m/s and is given by   v  =            d      x              d      t       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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