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Question

What is the multiplicative rate of change for the exponential function f(x) = f start bracket x end bracket equals two start bracket five-halves end bracket superscript negative x?

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Parker

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Solution By Steps

Step 1: Find the Derivative of the Exponential Function

To find the multiplicative rate of change, we need to differentiate the function f(x) with respect to x.

Step 2: Differentiate the Exponential Function

The derivative of the function f(x) = 2(5/2)^(-x) can be found using the chain rule and the derivative of a constant.

Step 3: Apply the Chain Rule

The derivative of 2(5/2)^(-x) is ln(5/2) * 2(5/2)^(-x).

Step 4: Simplify the Derivative

Simplify ln(5/2) * 2(5/2)^(-x) to get -ln(5/2) * 2(5/2)^(-x).

Final Answer

The multiplicative rate of change for the exponential function f(x) = 2(5/2)^(-x) is -ln(5/2) * 2(5/2)^(-x).

Summary

The multiplicative rate of change of an exponential function represents how the function’s output changes relative to its current value as the input changes. In this case, the rate of change is given by the derivative of the function. Understanding the rate of change is crucial in various fields such as finance, biology, and physics, where exponential growth or decay models are commonly used to describe real-world phenomena.

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