~In many problems, it is not necessary to derive it, just use the end result of the

derivation, i.e., the equation that results. Here is the derivation:

 

~It comes from the statement, "the rate of growth/decay of a quantity Q is proportional

to the amount of Q present at any given time t".  This translates to the following:

 

dQ/dt = kQ,  where k is the constant of the proportion. If k>0, there will be a growth, since

Q will increase with time. If k<0, there will be a decay, since Q will decrease with time.

 

~Separating the variables in this simple differential equation, we get:  dQ/Q =kdt.

 

~Integrating both sides,  ln(Q)=kt+C.  Assuming that the initial amount of Q is Qo (at t=0),

we get:   ln(Qo)=C.  Substituting back for C, we have:   ln(Q)=kt+ln(Qo)

 

~Getting the natural logs on the same side:   ln(Q)-ln(Qo)=kt  then using the division

property of logs (i.e., the log of a quotient is equivalent to subtracting the log of the denominator from the log of the numerator), we get;   ln(Q/Qo)=kt.  Now, use a property of the natural log to get:

 

Q/Qo = e^kt.  Multiplying both sides by Qo, we get the final result:   Q=Qoe^kt.

 

~This is the model used for simple growth/decay. You can start many problems here

and do not have to go through the derivation repeatedly.

~You will see this type of model in simple growth/decay problems dealing with population (either people or organisms) and with radioactive decay (see radiocarbon dating link)

~In all problems using this model, predictions are made, either for Q (amount of the quantity) at some value of time (t) or the time (t) is predicted at some value of Q.  In both cases, the constant k must be found.  In most cases, the initial amount of Q, which is Qo needs to be known (except in radioactive decay when the half-life is known).  

~Once these quantities are known, then there will be a direct tie between the two variables, Q and t.  Thus, if one is given, the other can be estimated.

~Let me emphasize that this model is for simple growth/decay and is not very realistic in many cases.  Other more advanced models are used in those cases (i.e., the logistic model for growth or for temperature decay, Newton's law of cooling).