Overview
A derivative of a function describes its rate of change at a particular point on the function. The rate of change doesn’t have to be constant, so it can be approximated along any point of a curve. Derivatives in calculus have many applications in quantitative sciences such as physics and chemistry.
Geometric Definition
Not all functions are linear. If a function is continuous, so that very small changes in input result in changes in output, the shape of the graph is a curve. In order to approximate the amount of change at any point on the curve, a tangent line can be dropped. The derivative is the measurement of the slope of the line at that point.
Differentiation
Differentiation is the process of finding derivatives. Both Newton and Leibniz used differentiation in the process of developing calculus. The differential is an infinitesimal change in a varying quantity, and can be related to all other changes in a function. Even though the change is infinitely small, it can still be measured by an approximation.
Derivatives in Space and Time
In order to measure changes in space and time, derivatives are used in differential equations. The time derivative, or rate of change over time, is significant to concepts such as velocity and acceleration. (It can be said that Newton developed calculus to quantify his observations in classical mechanics.) For example, velocity is the rate of change in position with respect to time. Acceleration is the rate of change of velocity over time. It is not necessarily constant, and can involve minute adjustments of speed.
Other Applications
Derivatives and differential equations are used in quantitative sciences and modeling. For example, the reaction rate in chemistry is a rate of change, measured by differential equations. Most measures of behavior (such as in psychology, sociology, and economics) can be approximated by the normal curve, which represents change that is continuous but not linear. Many high school students take standardized tests such as the SAT and the ACT. Changes in scores over time can be modeled using advanced statistical techniques, based upon derivatives and differential equations.
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