The use of exponential functions in everyday life

Exponential functions are an important class of mathematical functions. Generally they are defined as the solutions of the following differential equation:

\(\frac{df(t)}{dt} = C f(t)\),

where \(C\) is a constant and \(t\) is a variable, for example time. This means that exponential functions describe situations where the rate of change of something depends on how much of that something is currently available. The rate of change basically answers the question: How many new things do I get after a specific time?

An example is the spreading of a contagious disease. The number of people that get sick after a certain period of time, for example after one day, depends on how many people already have the disease.

Another example is the growth of money due to the interest rate. The amount of money that is added to my money after one year depends on how much money I already have in my bank account.