# Exponential growth

__Exponential growth__ is a specific way that a quantity may increase over time. It occurs when the instantaneous Rate (mathematics)#Of change (that is, the derivative) of a quantity with respect to time is proportionality (mathematics) to the quantity itself. Described as a Function (mathematics), a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth) - wikipedia

The graph illustrates how exponential growth (green) surpasses both linear (red) and cubic (blue) growth. - wikimedia

If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing exponential decay instead. In the case of a discrete Domain of a function of definition with equal intervals, it is also called __geometric growth__ or __geometric decay__ since the function values form a geometric progression.

The formula for exponential growth of a variable ''x'' at the growth rate ''r'', as time ''t'' goes on in discrete intervals (that is, at integer times0,1,2,3,...), is

:[itex]x_t = x_0(1+r)^t[/itex]

where ''x''<sub>0</sub> is the value of ''x'' at time 0. The growth of a bacterial Colony (biology) is often used to illustrate it. One bacterium splits itself into two, each of which splits itself resulting in four, then eight, 16, 32, and so on. The rate of increase keeps increasing because it is proportional to the ever-increasing number of bacteria. Growth like this is observed in real-life activity or phenomena, such as the spread of virus infection, the growth of debt due to compound interest, and the spread of viral videos. In real cases, initial exponential growth often do not last forever, slowing down due to upper limits caused by external factors and turning into the logistic curve.

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