Derivative
Introduction
In mathematics, the derivative is a fundamental concept in calculus, specifically in differential calculus. The derivative measures how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
Definition
The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time is advanced.
Notation
The derivative of a function f at a point is denoted in several ways. While f′(a) is common, it could also be denoted as df(a)/dx, Df(a), or D_x f(a), among others.
Mathematical Rigor
In the 17th century, Isaac Newton and Gottfried Wilhelm Leibniz brought a much more rigorous definition to the concept of the derivative. They were able to solve problems involving rates of change using infinitesimals, or quantities that are infinitely small.
Derivative Rules
There are several rules in calculus that allow for easier computation of most derivatives. The most common rules include the power rule, product rule, quotient rule, chain rule, and the rules for differentiating trigonometric functions.
Applications
The concept of a derivative is fundamental in many areas of mathematics and its applications. In physics, the derivative is used to determine the velocity and acceleration of moving objects. In biology, it is used to model populations and predict their behaviors. In finance, the derivative is used to model and predict changes in financial markets.