How compound interest works? Compound interest is a free charged for borrowing or depositing the money and you pay or earn a interest.
It is a percentage charged on the principle amount for a period of a year.
Formular:
P = The principle (The initial amount you borrow or deposit)
r = The annual rate of interest (The percentage)
t = The number of year the amount is deposited or borrowed
n = The number of times the interest is compounded per year
A = The amount after time t
 An amount of $1,000.00 is deposited in a bank paying an annual interest rate of 6%, compounded quarterly. Find the balance after 1 years. We know that P = 1000, r = 6/100 = 0.06, n = 4, and t = 1 Then put numbers in to formular 
We found out that no matter what is the numbers n is, the balance wil not exceed to 1061.84. In other word the balance will chang only to the nearest cent.
To prove this, we connect the concept of compound interest to the exponential function :
 Connecting the concept
 If we set h = r/n, and 1/h = n/r
so we get
 As the frequency of the compounding is increasing, m gets large and h approach to 0. To see what happen to the compound amount, we must therefore examine the limit.
 we know that:
 we can prove by subtitude number of h
 As the number get closer to 0 the result is the same as e.
We using this fact together with two limit theorems, we have
 so we can say that
 Example:
When $1000 is invested for one year at 6% interest compounded contunuously.
So we know that P = 1000, r = 0.06, and t = 1.
Then compound interest is
 Apply to calculus
One thousand dollars is invested at 5% interest compounded continuously
(a) Giv the formular A(t), the compound amount after t years.
(b) How much will be in the account after 6 years?
(c) After 6 years, at what rate will A(t)be growing?
(d) How long is required for the initial investment to double?
Solution
(a) P = 1000 and r = 0.05.
so
 (b)
 (c)
 (d)
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