Question

How does linear growth differ from exponential growth?

Answer 1

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.