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Concept Version 8
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Differentiation and Rates of Change in the Natural and Social Sciences

Differentiation, in essence calculating the rate of change, is important in all quantitative sciences.

Learning Objective

  • Give examples of differentiation, or rates of change, being used in a variety of academic disciplines


Key Points

    • Differentiation has applications to nearly all quantitative disciplines, whether it's natural or social science.
    • Physical scientists use differentiation and rate of change to study the way a physical concept changes over time or distance.
    • Social scientists use differentiation and rate to determine how people, goods, and processes change due to the change of an independent variable.

Terms

  • gross domestic product

    a measure of the economic production of a particular territory in financial capital terms over a specific time period

  • momentum

    (of a body in motion) the product of its mass and velocity

  • differential geometry

    the study of geometry using differential calculus


Full Text

Given a function $y=f(x)$, differentiation is a method for computing the rate at which a dependent output $y$ changes with respect to the change in the independent input $x$. This rate of change is called the derivative of $y$ with respect to $x$.

Rate of change is an important concept in many quantitative studies, and it is no surprise that differentiation (representing the rate of change) has applications to nearly all quantitative disciplines. For example, in physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. Newton's second law of motion states that the derivative of the momentum of a body equals the force applied to the body. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories.

Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory and abstract algebra.

Rates of change occur in all sciences and across all disciplines. Economists study the rate of change of gross domestic product and social scientists the rate in which populations vote in a specific area. Geologists study the rate of earth shift and the temperature gradient of rocks near a volcano. Accountants study the rate of change of production and supplies, and how any change can affect cost and profit. Urban engineers study the flow of traffic in order to design and build more efficient roads and freeways.

In every aspect of life in which something changes, differentiation and rates of change are an important aspect in understanding the world and finding ways to improve it.

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