Hey groggy,
I think what you are confused with is the different use between growth or inflation as used in Economics and in Physics.
In Economics, one usually reports the
relative growth of a certain quantity (GDP, overall prices, population, etc.) between two consecutive periods. Let the quantity we measure in the
nth period be denoted with
Pn. Then, the increase in the quantity between the
nth period and the (
n+1)-st period is simply given as the difference:
.
The relative increase is given by:
.
Sometimes we define the growth quotient:
.
Let us give a specific example. If someone says that the growth rate is 5%, it means that
rn = 0.05 and
qn = 1.05. Sometimes the growth can be negative (like in a recession). Then,
rn < 0 and
qn < 1.
If we know the values of
qn for every
n and the initial value
P0, then we can calculate
Pn:
So, if you had this in mind for the rate of change, then you were right:
However, this is not what is meant as a rate of change in Physics. To see this, we will take the continuum limit in the definition for
rn in the following manner. Let
T be the time between two periods. Then, if the initial instant
t0 = 0, we can take the instant when the
nth period begins to be
tn =
n*
T. Then, we may consider any discrete sequence {
xn} (such as
Pn or
rn) simply as the value of a continuous function
x(
t) at discrete time intervals. Then, if we formally let
, we will have the values of the function for all instants of time. Notice that:
so
So, the growth rate is really the
logarithmic derivative of a quantity (albeit taken at a discrete set of instants in time).
The acceleration is simply defined as the
ordinary derivative of the velocity: