What is the formula for the rate of change in an exponential function?

In the context of exponential functions, the correct formula that describes the behavior of an exponential growth or decay process is represented by ( Y = A(b)^t ), where ( A ) is the initial amount, ( b ) is the base of the exponential function representing the growth or decay factor, and ( t ) is typically the time variable.

This formula is foundational in expressing how quantities change exponentially. The base ( b ) indicates the rate of growth or decay: if ( b > 1 ), the function shows exponential growth, while if ( 0 < b < 1 ), the function represents exponential decay. The variable ( t ) commonly represents time or another incremental measure, reflecting how the quantity ( Y ) evolves as ( t ) changes.

In the context of rate of change specifically, this formula demonstrates how the output value ( Y ) increases or decreases in relation to changes in ( t ), effectively capturing the essence of exponential growth or decay. This makes it particularly relevant in real-world applications, such as population growth, radioactive decay, or interest calculations, where such relationships often manifest.