How to Calculate Instantaneous Rate

Calculus was invented by Sir Isaac Newton around 1600.
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Instantaneous rate of change is a concept at the core of basic calculus. It tells you how fast the value of a given function is changing at a specific instant, represented by the variable x. To find out how the quickly the function value changes, it’s necessary to find the derivative of the function, which is just another function based on the first. Inputting an x value into a function gives you a value. Inputting an x value into a derivative tells you how quickly that value is changing as x grows and shrinks.

    Determine your function. It will probably be given to you in the problem. For example, your function might be F(x) = x^3.

    Choose the instant (x value) you want to find the instantaneous rate of change for. For example, your x value could be 10.

    Derive the function from Step 1. For example, if your function is F(x) = x^3, then the derivative would be F’(x) = 3x^2.

    Input the instant from Step 2 into the derivative function from Step 3. F'(10) = 3x10^2 = 300. 300 is the instantaneous rate of change of the function x^3 at the instant 10.


    • If you need to know the rate of acceleration at a given instant instead of the rate of change, you should perform Step 3 twice in a row, finding the derivative of the derivative.


About the Author

L.P. Klages is an entrepreneur and software developer, concentrating on information theory, software user experience, and mathematical modeling. He has been writing about technology and the business of technology since 1999. His articles have appeared on many sites, including,, and eHow. Klages attended Jacksonville University in Jacksonville, Fla.

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