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Line 1: − A '''derivative''' is the result of [[differentiation]] - a measure in [[Calculus]] of how functions change based on how their input values change; otherwise known as 'the rate of change' (second derivatives thus give the rate of change of change, etc). For example, [[velocity]] is obtained from taking the first [[derivative]] of a [[position]] [[function]] with respect to time. [[Acceleration]] can be obtained either by taking the [[derivative]] of a [[velocity]] [[function]] with respect to time or by taking the time [[derivative]] of a [[position]] [[function]] twice. Note that when using this method, taking the derivative the first time yields the [[velocity]] function, and taking the derivative the second time yields the [[acceleration]] function.+ − The terms [[speed]] and [[velocity]] are often used interchangeably, which is incorrect. [[Speed]] is a [[Scalar quantity]] which refers to the magnitude, or size, of the [[velocity]] [[vector quantity|vector]]. [[Velocity]] has a direction, while [[speed]] does not.+ − Given a graph of a [[real numbers|real]] curve, the derivative at a specific point will equal the [[slope]] of the line + − [[tangent]] to that point. − To calculate the derivative of a function, one must use techniques from the differential branch of calculus. This branch of calculus is related to the integral branch by the first fundamental theorem of calculus: ''differentiation'' (the process of finding a derivative) ''is the reverse process of [[integral|integration]]'' (the process of finding an integral).+ − + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + − +
Combined two articles and wrote some more
In mathematics a '''derivative''' is measure of how functions change. Algebraic '''differentiation''' is an important part of [[calculus]], an essential branch of [[mathematics]] in the modern age. Differentiation can be used, for example, in [[mechanics]] to find the acceleration of an object from a velocity-time graph.
Essentially, differentiation is employed as a means to calculate the [[gradient]] or rate of change at a particular value for a given function, ''f''. Consequently, it can be used to calculate velocity from a displacement-time graph, or acceleration from a velocity-time graph. Furthermore, it can be used to calculate the rate of cooling from a temperature-time graph. These are a few examples of the applications of differentiation.
When defined from the first principals, the derivative of a function is the gradient of a function over <math>[x,x+h]</math>. If <math>h</math> is allowed to appoach 0 then the gradient approches the gradient at the point <math>x</math>:
:<math>f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}</math>
In mathematics, derivatives are helpful in determining the [[maximum|maxima]] and [[minimum|minima]] of a function. For example, taking the derivative of a quadratic function will yield a linear function. The points at which this function equals zero are called ''critical'' points. Maxima and minima can occur at critical points, and can be verified to be a maximum or minimum by the ''second derivative test''. The second derivative is used to determine the [[concavity]], or curved shape of the graph. Where the concavity is positive, the graph curves upwards, and could contain a relative minimum. Where the concavity is negative, the graph curves downwards, and could contain a relative maximum. Where the concavity equals zero is said to be a point of ''inflection,'' meaning that it is a point where the concavity could be changing. Also, differentials have numerous applications in physics.
Alternative notation also commonly found is <math>y=f(x)</math> and <math>\frac{dy}{dx}=f'(x)</math>.
In [[classical mathematics]], a function can be be differentiated using the general formula:
:<math>\frac{dy}{dx} = nx^{(n-1)}</math>
(Proving this is a worth while exercise).
For example, if <math>y = 3 x^2</math>, the derivative with respect to <math>x</math> is
:<math>\frac{dy}{dx} = 6 x</math>
Thus the derivative is a measurement of how a function changes when the values of its inputs vary. Derivatives are helpful in determining the [[maximum|maxima]] and [[minimum|minima]] of a function. For example, taking the derivative of a quadratic function will yield a linear function. The points at which this function equals zero are called ''critical'' points. Maxima and minima can occur at critical points, and can be verified to be a maximum or minimum by the ''second derivative test''. The second derivative is used to determine the [[concavity]], or curved shape of the graph. Where the concavity is positive, the graph curves upwards, and could contain a relative minimum. Where the concavity is negative, the graph curves downwards, and could contain a relative maximum. Where the concavity equals zero is said to be a point of ''inflection,'' meaning that it is a point where the concavity could be changing. Also, differentials have numerous applications in physics.
==Properties of the derivative==
*<math>\frac{d}{dx}(f(x)+g(x))=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)</math>
*<math>\frac{d}{dx}cf(x)=c\frac{d}{dx}f(x)</math>
*<math>\frac{dy}{dx}=\frac{dy}{dt}\cdot\frac{dt}{dx}</math>
<math>\frac{dy}{dx}</math> is a proper quotient and a result,
:<math>\frac{dy}{dx}=f(x) \Leftrightarrow dy=f(x)dx</math>
is a valid operation and is much used in solving [[differential equations]].
The differential operator has an associated [[eigenfunction]],
:<math>\frac{dy}{dx}=y\Leftrightarrow y=e^{x}</math>
where [[e]] is the constant defined for this purpose.
===Important differentiation rules===
*[[Product rule]]
*[[Quotient rule]]
*[[Chain rule]]
The roots of differentiation are profoundly linked with tangency; ergo, this aspect of mathematics can first be perceived to have been developed during the time of the [[Ancient Greeks]] through the work of Greek geometers like [[Euclid]], [[Sanath]] and [[Archimides]].
[[Category:Calculus]]
[[Category:Calculus]]
[[Category:Mathematics]]
[[Category: Mathematics]]