Everything about Derivative totally explained
In
calculus, a branch of
mathematics, the
derivative is a measurement of how a
function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point. For example, the derivative of the position or distance of a car at some point in time is the instantaneous velocity, or instantaneous speed (respectively), at which that car is traveling (conversely the
integral of the velocity is the car's position).
A closely related notion is the
differential of a function.
The derivative of a function at a chosen input value describes the best
linear approximation of the function near that input value. For a
real-valued function of a single real variable, the derivative at a point equals the
slope of the
tangent line to the
graph of the function at that point. In higher dimensions, the derivative of a function at a point is a
linear transformation called the
linearization.
The process of finding a derivative is called
differentiation. The
fundamental theorem of calculus states that differentiation is the reverse process to
integration.
Differentiation and the derivative
Differentiation is a method to compute the rate at which a quantity,
y, changes with respect to the change in another quantity,
x, upon which it's
dependent. This rate of change is called the
derivative of
y with respect to
x. In more precise language, the dependency of
y on
x means that
y is a
function of
x. If
x and
y are
real numbers, and if the
graph of
y is plotted against
x, the derivative measures the
slope of this graph at each point. This functional relationship is often denoted
y =
f(
x), where
f denotes the function.
The simplest case is when
y is a
linear function of
x, meaning that the graph of
y against
x is a straight line. In this case,
y =
f(
x) =
m x +
c, for real numbers
m and
c, and the slope
m is given by
»
The existence of the Jacobian is strictly stronger than existence of all the partial derivatives, but if the partial derivatives exist and satisfy mild smoothness conditions, then the total derivative exists and is given by the Jacobian.
The definition of the total derivative subsumes the definition of the derivative in one variable. In this case, the total derivative exists if and only if the usual derivative exists. The Jacobian matrix reduces to a 1×1 matrix whose only entry is the derivative
f′ (x). This 1×1 matrix satisfies the property that
f(
a +
h) −
f(
a) −
f′(
a)
h is approximately zero, in other words that
»
Up to changing variables, this is the statement that the function
is the best linear approximation to
f at
a.
The total derivative of a function doesn't give another function in the same way that one-variable case. This is because the total derivative of a multivariable function has to record much more information than the derivative of a single-variable function. Instead, the total derivative gives a function from the
tangent bundle of the source to the tangent bundle of the target.
Generalizations
The concept of a derivative can be extended to many other settings. The common thread is that the derivative of a function at a point serves as a
linear approximation of the function at that point.
An important generalization of the derivative concerns complex functions of complex variables, such as functions from (a domain in) the complex numbers C to C. The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition. However, this innocent definition hides some very deep properties. If C is identified with R² by writing a complex number z as x + i y, then a differentiable function from C to C is certainly differentiable as a function from R² to R² (in the sense that its partial derivatives all exist), but the converse isn't true in general: the complex derivative only exists if the real derivative is complex linear and this imposes relations between the partial derivatives called the Cauchy Riemann equations — see holomorphic functions.
Another generalization concerns functions between differentiable or smooth manifolds. Intuitively speaking such a manifold M is a space which can be approximated near each point x by a vector space called its tangent space: the prototypical example is a smooth surface in R³. The derivative (or differential) of a (differentiable) map f: M → N between manifolds, at a point x in M, is then a linear map from the tangent space of M at x to the tangent space of N at f(x). The derivative function becomes a map between the tangent bundles of M and N. This definition is fundamental in differential geometry and has many uses — see pushforward (differential) and pullback (differential geometry).
Differentiation can also be defined for maps between infinite dimensional vector spaces such as Banach spaces and Fréchet spaces. There is a generalization both of the directional derivative, called the Gâteaux derivative, and of the differential, called the Fréchet derivative.
One deficiency of the classical derivative is that not very many functions are differentiable. Nevertheless, there's a way of extending the notion of the derivative so that all continuous functions and many other functions can be differentiated using a concept known as the weak derivative. The idea is to embed the continuous functions in a larger space called the space of distributions and only require that a function is differentiable "on average".
The properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology — see, for example, differential algebra.
Also see arithmetic derivative.Further Information
Get more info on 'Derivative'.
|
External Link Exchanges
Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:
<a href="http://derivative.totallyexplained.com">Derivative Totally Explained</a>
Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned. |
