How does linear growth differ from exponential growth?

1 Answer
Aug 6, 2014

Linear growth is always at the same rate, whereas exponential growth increases in speed over time.

A linear function like #f(x)=x# has a derivative of #f'(x)=1#, which means that it has a constant growth rate. No matter how long the object or population is growing, no matter what its size, the growth rate will always be #1# - no exceptions.

On the other hand, an exponential function like #g(x)=e^x# has a derivative of #g'(x)=e^x#. This means that as #x# gets larger, the derivative also increases along with it - meaning that the graph gets steeper and the growth rate gets faster. In fact, the growth rate continues to increase forever.