A derivative is an object that is based on, or created from, a basic or primary source. This meaning is particularly important in linguistics and etymology, where a derivative is a word that is formed from a more basic word. Similarly in chemistry a derivative is a compound that is formed from a similar compound.
In finance, derivative is the common short form for derivative security.
In mathematics, the derivative of a function is one of the two central concepts of calculus. The inverse of a derivative is called the antiderivative, or indefinite integral.
The derivative of a function at a certain point is a measure of the rate at which that function is changing as an argument undergoes change. A derivative is the computation of the instantaneous slopes of f(x) at every point x. This corresponds to the slopes of the tangents to the graph of said function at said point; the slopes of such tangents can be approximated by a secant. Derivatives can also be used to compute concavity.
Functions do not have derivatives at points where they have either a vertical tangent or a discontinuity.
Differentiation and Differentiability
Differentiation can be used to determine the change which something undergoes as a result of something else changing, if a mathematical relationship between two objects has been determined. The derivative of f(x) is written in several possible ways: f'(x) (pronounced f prime of x), d/dx[f(x)] (pronounced d by d x of f of x), df/dx (pronounced d f by d x), or D_{x}[f] (pronounced d sub x of f). The last three symbolisms are useful in considering differentiation as an operator, and these symbolisms are known as the differential operator.
A function is differentiable at a point x if its derivative exists at this point; a function is differentiable in an interval if a derivative exists for every x within the interval. If a function is not continuous at c, then there is no slope and the function is therefore not differentiable at c; however, even if a function is continuous at c, it may not be differentiable.
Newton's Difference Quotient
Derivatives are defined by taking the limit of a secant slope, as its two points of intersection (with f(x)) converge; the secant approaches a tangent. This is expressed by Newton's difference quotient; where h is Δx (the distance between the x-coordinates of the secant's points of intersection):Since immediately substituting 0, for h, results in division by zero, the numerator must be simplified such that h can be factored out and then canceled against the denominator. The resulting function, f '(x), is the derivative of f(x).
Click here for some examples of how to use this quotient.
The Alternative Difference Quotient
Above, the derivative of f(x) (as defined by Newton) was described as the limit, as h approaches zero, of [f(x + h) - f(x)] / h. An alternative explanation of the derivative can be derived from Newton's quotient. Using the above; the derivative, at c, equals the limit, as h approaches zero, of [f(c + h) - f(c)] / h; if one then lets h = x - c (and c + h = x); then, x approaches c (as h approaches zero); thus, the derivative equals the limit, as x approaches c, of [f(x) - f(c)] / (x - c). This definition is used for a partial proof of the Chain Rule.Stationary Points
Points on the graph of a function where the derivative equals zero are called "stationary points". If the second derivative is positive at a stationary point, that point is a local minimum; if negative, it is a local maximum; if zero, it may or may not be a local minimum or local maximum. Taking derivatives and solving for stationary points is often a simple way to find local minima or maxima, which can be useful in optimization.Notable Derivatives
- For logarithmic functions:
- For trigonometric functions
- The derivative of sinx, is cosx.
- The derivative of cosx, is -sinx.
- The derivative of tanx, is sec^{2}x.
- The derivative of cotx, is -csc^{2}x.
- The derivative of secx, is (secx)(tanx).
- The derivative of cscx, is -(cscx)(cotx).
Multiple Derivatives
When the derivative of a function of x has been found, the result, being also a function of x, may be also differentiated, which gives the derivative of the derivative, or second derivative. Similarly, the derivative of the second derivative is called the third derivative, and so on. One might refer to subsequent derivatives of f by:In order to avoid such "cumbersome" notation, the following options are often preferred:
Physics
Arguably the most important application of calculus, to physics, is the concept of the "time derivative" -- the rate of change over time -- which is required for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics:- Velocity (instantaneous velocity; the concept of average velocity predates calculus) is the derivative (with repsect to time) of an object's position.
- Acceleration is the derivative (with respect to time) of an object's velocity.
- Jerk is the derivative (with respect to time) of an object's acceleration.
For example, if an object's position ; then, the object's velocity is ; the object's acceleration is ; and the object's jerk is .
If the velocity of a car is given, as a function of time; then, the derivative of said function with respect to time describes the acceleration of said car, as a function of time.
Algebraic Manipulation
"Messy" limit calculations can be avoided, in certain cases, because of differentiation rules which allow one to find derivatives via algebraic manipulation; rather than by direct application of Newton's difference quotient. One should not infer that the definition of derivatives, in terms of limits, is unnecessary. Rather, that definition is the means of proving the following "powerful differentiation rules"; these rules are derived from the difference quotient.
- Constant Rule: The derivative of any constant is zero.
- Constant Multiple Rule: If c is some real number; then, the derivative of equals c multiplied by the derivative of f(x) (a consequence of linearity below)
- Linearity: for all functions f and g and all real numbers a and b.
- General Power Rule (Polynomial rule): If , for some real number r; .
- Product Rule: for all functions f and g.
- Quotient Rule: if .
- Chain Rule: If , then
- Inverse functions and differentiation: If , , and f(x) and its inverse are differentiable, then for cases in which when ,
- Derivative of one variable with respect to another when both are functions of a third variable: Let and . Now
- Implicit differentiation: If be an implicit function, we have: dy/dx = - (∂f / ∂x) / (∂f / ∂y).
As an example, the derivative of is .
Using Derivatives to Graph Functions
Derivatives are a useful tool for examining the graphs of functions. In particular, the points in the interior of the domain of a real-valued function which take that function to local extrema will all have a first derivative of zero. However, not all "critical points" (points at which the derivative of the function has determinant zero) are mapped to local extrema; some are so-called "saddle points". The Second Derivative Test is one way to evaluate critical points: if the second derivative of the function at the critical point is positive, then the point is a local minimum; if it is negative, the point is a local maximum; if it is neither, the point is either saddle point or part of a locally flat area (possibly still a local extremum, but not absolutely so). (In the case of multidimensional domains, the function will have a partial derivative of zero with respect to each dimension, at local extrema.)Once the local extrema have been found, it is usually rather easy to get a rough idea of the general graph of the function, since (in the single-dimensional domain case) it will be uniformly increasing or decreasing except at critical points, and hence (assuming it is continuous) will have values in between its values at the critical points on either side. Also, the supremum of a continuous function on an open and bounded domain will also be one of the local maxima; the infemum will be one of the local minima--this gives one an easy way to find the bounds of the function's range.
More Info
Where a function depends on more than one variable, the concept of a partial derivative is used. Partial derivatives can be thought of informally as taking the derivative of the function with all but one variable held temporarily constant near a point. Partial derivatives are represented as ∂/∂x (where ∂ is a rounded 'd' known as the 'partial derivative symbol'). Mathematicians tend to speak the partial derivative symbol as 'der' rather than the 'dee' used for the standard derivative symbol, 'd'.The concept of derivative can be extended to more general settings. The common thread is that the derivative at a point serves as a linear approximation of the function at that point. Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles.
In order to differentiate all continuous functions and much more, one defines the concept of distribution.
For differentiation of complex functions of a complex variable see also Holomorphic function.
See also: differintegral.
References
- Calculus of a Single Variable: Early Transcendental Functions (3rd Edition) by Edwards, Hostetler, and Larson (2003)\n