Difference between revisions of "Derivative (calculus)"

Revision as of 21:49, July 7, 2008

A derivative is a measure in Calculus of how functions change based on how their input values change. Given a graph of a 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 (the process of finding an integral).

In mathematics, derivatives are helpful in determining the maximum and minimum 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.

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-In mathematics the '''derivative''' is the rate of change of a function with respect to one of its variables. For example, if <math>r(t)</math> denotes the position of a particle on a line as a function of time, the derivative of <math>r</math> with respect to the time variable <math>t</math> is the rate of change of the particle's position with respect to time, i.e., the speed of the particle. Similarly, the derivative of the particle's speed with respect to time corresponds to the particle's acceleration. Other examples of derivatives in physics abound: for example, the derivative can be used to calculate the rate of cooling from a temperature-time graph.
+A '''derivative''' is a measure in [[Calculus]] of how functions change based on how their input values change. Given a graph of a [[real]] curve, the derivative at a specific point will equal the [[slope]] of the line [[tangent]] to that point.
-The process of finding a derivative is called '''differentiation'''.
+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 [[integration|integral]]'' (the process of finding an integral).
-Differentiation is one of two fundamental operations in [[calculus]], the other being [[integration]].
+In mathematics, derivatives are helpful in determining the [[maxima|maximum]] and [[minima|minimum]] 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.
-For a real-valued function <math>f(x)</math> of one variable, the derivative of <math>f</math> corresponds to the [[slope]] of the tangent line to the graph of ''f'' at ''x''.
-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]] {{fact}}.
-When defined from the first principals, the derivative of a function is the limit of the average rate of change of the function over the interval <math>[x,x+h]</math> as <math>h</math> tends to zero. In other words, the derivative
-:<math>f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}</math>
-provided the above limit exists.
-Alternative notation also commonly found is <math>\frac{df}{dx}=f'(x)</math>.
-For example, a polynomial can be differentiated by taking into account the linearity of the derivative, and by using the general formula:
-:<math>\frac{d}{dx}(x^n) = nx^{(n-1)}</math>
-(Proving this is a worth while exercise).
-For example, if <math>y = 3 x^2+2x</math>, the derivative with respect to <math>x</math> is
-:<math>\frac{dy}{dx} = 6 x+2</math>
-==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}=\lambda y\Leftrightarrow y=Ce^{\lambda x}</math>
-where [[e]] is the constant defined for this purpose.
-===Important differentiation rules===
-*[[Product rule]]
-*[[Quotient rule]]
-*[[Chain rule]]
-==Higher order derivatives==
-A higher order derivative is obtained by repeating derivatives,
-:<math>\frac{d^{2}y}{dx^{2}}=\frac{d}{dx}\left(\frac{dy}{dx}\right)</math>
-:<math>\frac{d^{3}y}{dx^{3}}=\frac{d}{dx}\left(\frac{d^{2}y}{dx^{2}}\right)=\frac{d}{dx}\left(\frac{d}{dx}\left(\frac{dy}{dx}\right)\right)</math>
-and so forth. The derivatives are termed the n-th order derivative e.g, second order derivative, third order derivative.
-A common alternative notation is,
-:<math>\frac{d^{2}y}{dx^{2}}=f''(x)</math>
-:<math>\frac{d^{3}y}{dx^{3}}=f'''(x)</math>
-:<math>\frac{d^{n}y}{dx^{n}}=f^{n}(x)</math>
-The dashes are actually roman numerals: the symbol for the forth order derivative would be a slanted IV.
-Continuing our earlier example; <math>y = 3 x^2+2x</math>, the second derivative with respect to <math>x</math> is
-:<math>\frac{d^{2}y}{dx^{2}} = \frac{d}{dx}(6 x+2)=6</math>
-:<math>\frac{d^{3}y}{dx^{3}} = \frac{d}{dx}(6)=0</math>
-All subsequent derivatives would be zero. For any [[polynomial]] of order n-1, <math>P_{n-1}(x)</math>,
-:<math>\frac{d^{n}}{dx^{n}}P_{n-1}(x)=0</math>
-Other than the second derivative these have no real applications{{fact}}. A function which has derivatives of all orders is called a [[smooth function]].
-==See Also==
-*[[Partial derivatives]]
-*[[Applications of derivatives]]
-[[Category:Calculus]]
-[[Category: Mathematics]]

Difference between revisions of "Derivative (calculus)"

Revision as of 21:49, July 7, 2008

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