Derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. The derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation.
There are multiple different notations for differentiation. Leibniz notation, named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas prime notation is written by adding a prime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leibniz notation by adding superscripts to the differentials, and in prime notation by adding additional prime marks. Higher order derivatives are used in physics; for example, the first derivative with respect to time of the position of a moving object is its velocity, and the second derivative is its acceleration.
Derivatives can be generalized to functions of several real variables. In this case, the derivative is reinterpreted as a linear transformation whose graph is the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.
Definition
As a limit
A function of a real variable is differentiable at a point of its domain, if its domain contains an open interval containing, and the limitexists. This means that, for every positive real number, there exists a positive real number such that, for every such that and then is defined, and
where the vertical bars denote the absolute value. This is an example of the -definition of limit.
If the function is differentiable at, that is if the limit exists, then this limit is called the derivative of at. Multiple notations for the derivative exist. The derivative of at can be denoted, read as " prime of "; or it can be denoted, read as "the derivative of with respect to at " or " by at ". See below. If is a function that has a derivative at every point in its domain, then a function can be defined by mapping every point to the value of the derivative of at. This function is written and is called the derivative function or the derivative of. The function sometimes has a derivative at most, but not all, points of its domain. The function whose value at equals whenever is defined and elsewhere is undefined is also called the derivative of. It is still a function, but its domain may be smaller than the domain of.
For example, let be the squaring function:. Then the quotient in the definition of the derivative is
The division in the last step is valid as long as. The closer is to, the closer this expression becomes to the value. The limit exists, and for every input the limit is. So, the derivative of the squaring function is the doubling function:.
The ratio in the definition of the derivative is the slope of the line through two points on the graph of the function, specifically the points and. As is made smaller, these points grow closer together, and the slope of this line approaches the limiting value, the slope of the tangent to the graph of at. In other words, the derivative is the slope of the tangent.
Using infinitesimals
One way to think of the derivative is as the ratio of an infinitesimal change in the output of the function to an infinitesimal change in its input. In order to make this intuition rigorous, a system of rules for manipulating infinitesimal quantities is required. The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals are an extension of the real numbers that contain numbers greater than anything of the form for any finite number of terms. Such numbers are infinite, and their reciprocals are infinitesimals. The application of hyperreal numbers to the foundations of calculus is called nonstandard analysis. This provides a way to define the basic concepts of calculus such as the derivative and integral in terms of infinitesimals, thereby giving a precise meaning to the in the Leibniz notation. Thus, the derivative of becomes for an arbitrary infinitesimal, where denotes the standard part function, which "rounds off" each finite hyperreal to the nearest real. Taking the squaring function as an example again,Continuity and differentiability
If is differentiable at, then must also be continuous at. As an example, choose a point and let be the step function that returns the value 1 for all less than, and returns a different value 10 for all greater than or equal to. The function cannot have a derivative at. If is negative, then is on the low part of the step, so the secant line from to is very steep; as tends to zero, the slope tends to infinity. If is positive, then is on the high part of the step, so the secant line from to has slope zero. Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist. However, even if a function is continuous at a point, it may not be differentiable there. For example, the absolute value function given by is continuous at, but it is not differentiable there. If is positive, then the slope of the secant line from 0 to is one; if is negative, then the slope of the secant line from to is. This can be seen graphically as a "kink" or a "cusp" in the graph at. Even a function with a smooth graph is not differentiable at a point where its tangent is vertical: For instance, the function given by is not differentiable at. In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative.Most functions that occur in practice have derivatives at all points or almost every point. Early in the history of calculus, many mathematicians assumed that a continuous function was differentiable at most points. Under mild conditions, this is true. However, in 1872, Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere. This example is now known as the Weierstrass function. In 1931, Stefan Banach proved that the set of functions that have a derivative at some point is a meager set in the space of all continuous functions. Informally, this means that hardly any random continuous functions have a derivative at even one point.
Notation
One common way of writing the derivative of a function is Leibniz notation, introduced by Gottfried Wilhelm Leibniz in 1675, which denotes a derivative as the quotient of two differentials, such as and. It is still commonly used when the equation is viewed as a functional relationship between dependent and independent variables. The first derivative is denoted by, read as "the derivative of with respect to ". This derivative can alternately be treated as the application of a differential operator to a function, Higher derivatives are expressed using the notation for the -th derivative of. These are abbreviations for multiple applications of the derivative operator; for example, Unlike some alternatives, Leibniz notation involves explicit specification of the variable for differentiation, in the denominator, which removes ambiguity when working with multiple interrelated quantities. The derivative of a composed function can be expressed using the chain rule: if and thenAnother common notation for differentiation is by using the prime mark in the symbol of a function. This notation, due to Joseph-Louis Lagrange, is now known as prime notation. The first derivative is written as, read as " prime of ", or, read as " prime". Similarly, the second and the third derivatives can be written as and, respectively. For denoting the number of higher derivatives beyond this point, some authors use Roman numerals in superscript, whereas others place the number in parentheses, such as or. The latter notation generalizes to yield the notation for the th derivative of.
In Newton's notation or the dot notation, a dot is placed over a symbol to represent a time derivative. If is a function of, then the first and second derivatives can be written as and, respectively. This notation is used exclusively for derivatives with respect to time or arc length. It is typically used in differential equations in physics and differential geometry. However, the dot notation becomes unmanageable for high-order derivatives and cannot deal with multiple independent variables.
Another notation is D-notation, which represents the differential operator by the symbol. The first derivative is written and higher derivatives are written with a superscript, so the -th derivative is. This notation is sometimes called Euler notation, although it seems that Leonhard Euler did not use it, and the notation was introduced by Louis François Antoine Arbogast. To indicate a partial derivative, the variable differentiated by is indicated with a subscript, for example given the function, its partial derivative with respect to can be written or. Higher partial derivatives can be indicated by superscripts or multiple subscripts, e.g. and.