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1. Mathematical functions :
Differentiation is a process of looking at the way of function changes from one
point to another. In particular ,it measures how rapidly a function is changing at an
point. Differentiation is a method of finding the derivative of a function.
Differentiation is a process in maths ,where we find one instantaneous rate of
change in function based on one of it’s variabels. values of particular functions

2. ANALYSIS :
We consider the growth of a population (for example, human, an animal, or a bacteria colony) as a
function of time t. Let x (t) be the size of the population at any time t .
Assume that population grows at a rate directly proportional to the amount of population present at that
time. Then, we obtain dx/dt = k(x),
Solution of the differential equation is x(t)=Ce^kt, where C is a constant of integration. The values of C
and k are determined with the help of initial conditions. Thus, the population increases exponentially with
time.

3. CALCULATION :
The growth of a population is proportional to the number present. If the population of a colony
doubles in 30 years, in how many years will the population become triple?
Solution :
Let x(t) be the population at time t . Then dx/dt = kx. By separating the variables, we
obtain dx/x = kdt.
Integrating on both sides, we get, log |x| = kt + log |C| or x = Ce^kt, where C is an
arbitrary constant.
Let x0 be the population when t = 0 and obtain C = x0 .
Thus, we get x = x0e^kt
Now x = 2x0 , when t = 30 and thus, k = 1/30(log 2).
Hence, x = x0 2^(t/30) is the population at any time t.
Assume that the population is tripled in t1 years.
That is, x = 3x0 , when t = t1 . Thus, t1= 30 (log3/log2).
Therefore, the population is tripled in 30 (log3/log2) year. (or) 47.5 years.

4. RESULT AND DISCUSSION:
The subject of differential equations has vast applications in solving real
world problems. The solutions of the differential equations are used to
predict the behaviours of the system at a future time, or at an unknown
location. So differentiation is use to predict the future population of
particular area.

5. ADVANTAGES
Able to predict the future population our country using python program and determine
the time of death. This project can help to measure and manage population growth in
the current world setup.
DISADVANTAGES
It is only for the prediction not for exact solution.It is hard to predict the future population and time
consuming.

6. CONCLUSION :
Using the techniques above I have been able to predict the population size of our
country. some of the method displayed above to improve their social and
economic status.