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

*n*th period be denoted with

*P*_{n}. Then, the increase in the quantity between the

*n*th 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

*r*_{n} = 0.05 and

*q*_{n} = 1.05. Sometimes the growth can be negative (like in a recession). Then,

*r*_{n} < 0 and

*q*_{n} < 1.

If we know the values of

*q*_{n} for every

*n* and the initial value

*P*_{0}, then we can calculate

*P*_{n}:

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

*r*_{n} in the following manner. Let

*T* be the time between two periods. Then, if the initial instant

*t*_{0} = 0, we can take the instant when the

*n*th period begins to be

*t*_{n} =

*n**

*T*. Then, we may consider any discrete sequence {

*x*_{n}} (such as

*P*_{n} or

*r*_{n}) 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: