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Topic: Inverse functions and differentiation

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	Inverse functions and differentiation - Wikipedia, the free encyclopedia
		In mathematics, the inverse of a function y = f(x) is a function that, in some fashion, "undoes" the effect of f (see inverse function for a formal and detailed definition).
		Differentiation in calculus is the process of obtaining a derivative.
		It follows that functions with continuous derivative have inverses in a neighbourhood of every point where the derivative is non-zero.
	en.wikipedia.org /wiki/Inverse_functions_and_differentiation (269 words)

	Derivative - Wikipedia, the free encyclopedia
		Points on the graph of a function where the derivative is undefined or equals zero are called critical points or sometimes stationary points (in the case where the derivative equals zero).
		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.
		For complex functions of a complex variable differentiability is a much stronger condition than that the real and imaginary part of the function are differentiable with respect to the real and imaginary part of the argument.
	en.wikipedia.org /wiki/Derivative (2201 words)

	Inverse - Wikipedia, the free encyclopedia
		Inverse multiplexer - Splits a signal into several signals, opposite of a multiplexer.
		Inverse perspective - Also Byzantine perspective: the further the objects, the larger they are drawn.
		Inverse-square law - The magnitude of a force is proportional to the inverse square of the distance.
	en.wikipedia.org /wiki/Inverse (304 words)

	Encyclopedia: Inverse functions and differentiation
		In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range).
		In mathematics, an inverse function is in simple terms a function which does the reverse of a given function.
		In mathematics, the inverse function theorem gives sufficient conditions for a vector-valued function to be invertible on an open region containing a point in its domain.
	www.nationmaster.com /encyclopedia/Inverse-functions-and-differentiation (702 words)

	Derivative - Open Encyclopedia (Site not responding. Last check: 2007-09-17)
		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.
		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.
		The simplest notation for differentiation that is in current use is due to Lagrange and uses the prime, ′.
	open-encyclopedia.com /Derivative (2077 words)

	Talk:Inverse functions and differentiation - Wikipedia, the free encyclopedia
		The idea is to get to the point as fast as possible, point people who want to know more about inverse functions and calculus at the right places, and then gives some examples not just of inverse functions but of the whole reciprocal thing.
		Examples of the derivatives of inverse functions were under development(by pizza), and there was a perfect spot for them at the bottom.
		The difference between continuous and differentiable, and differentiable, is slight, especially given the fact that were a function is not continuous, it is not differentiable.(however, the converse does not always hold) In any case, this is quite a small point, which could be easily 'tweaked' by a very very minor correction.
	www.wikipedia.org /wiki/Talk:Inverse_functions_and_differentiation (1773 words)

	Derivatives of Inverse Functions
		A first step in the proof is to show that the inverse of a continuous function is continuous; this proof in turn requires an application of the Intermediate Value Theorem (see Stage 4) and actually reveals a fairly deep and subtle property of the real numbers.
		The square root function is the inverse of the squaring function f(x)=x
		The differentiability theorem for inverse functions guarantees that the square root function is differentiable at x whenever f '(x)=2x is not equal to zero.
	oregonstate.edu /instruct/mth251/cq/Stage6/Lesson/inverseDeriv.html (615 words)

	Derivatives of Inverse Trigonometric Functions
		However, it is generally enought to consider the inverse sine and the inverse tangent functions.
		We apply the chain rule to the left end, remembering that the derivative of the sine function is the cosine function and that y is a differentiable function of x.
		Remember, when we differentiate a function of x in terms of x (this is the meaning of the dx in d/dx), we must express our answer in terms of x.
	oregonstate.edu /instruct/mth251/cq/Stage6/Lesson/invTrigDeriv.html (731 words)

	Derivative - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-09-17)
		Points on the graph of a function where the derivative equals zero are called critical points or sometimes stationary points.
		Inverse functions and differentiation: If,, and f(x) and its inverse are differentiable, then for cases in which when,
		For differentiation of complex functions of a complex variable see also Holomorphic function.
	encyclopedia.learnthis.info /d/de/derivative.html (1805 words)

	Inverse Functions; Chain Rule; Implicit Differentiation
		We note that this function can be simplified using the laws of exponents to obtain e ^ ln x * e^1 = x * e^2 = e * x; since e is a constant number the derivative will simply be e * 1 = e.
		Its inverse function is f(x) = sin(x), and as usual the composite function f(g(x)) = x with derivative 1.
		Since F is given as a function of v and v is given as a function of t, we use the chain rule in form dF / dt = dF / dv * dv / dt and we obtain dF / dt = 48,000 v * (.006 t).
	www.vhcc.edu /cal1fall/lectures/ca1_981102_981113/ca1_1104/class_notes.htm (1646 words)

	Solving differential equations
		function y = f(x) is a function that, in some fashion, "undoes" the effect of f (see
		inverse function for a formal and detailed definition).
		Geometrically, a function and inverse function have graphs that are
	math-tables.net /inverse.html (168 words)

	Inverse Functions, Part 3
		In Part 2 we constructed the inverse of the (restricted) sine function as a "function defined by an integral":
		The inverse sine function is, by definition, the inverse of the function defined by that equation (with the domain restriction already noted).
		The inverse of the tangent function (arctangent, denoted arctan x) satisfies the equation tan y = x, where x is the independent variable, and y is the dependent variable.
	www.math.duke.edu /education/ccp/materials/intcalc/inverse/inverse3.html (628 words)

	Inverse functions and differentiation: Definition and Links by Encyclopedian.com - All about Inverse functions and ...
		Inverse functions and differentiation: Definition and Links by Encyclopedian.com - All about Inverse functions and differentiation
		$\frac{dy}{dx}$ denotes the derivative of the function $y=f(x)$ with respect to $x$ .
		$\frac{dx}{dy}$ denotes the derivative of the function $x=f(y)$ with respect to $y$ .
	www.encyclopedian.com /in/Inverse-functions-and-differentiation.html (262 words)

	Derivatives Of Trig Functions -
		relationship between the derivatives of trig functions and their cofunctions, the relationship between the derivatives of hyperbolic trig functions and their reciprocal functions...
		Derivatives of Trig Functions, Exponential Functions, Logarithm Functions, Inverse Trig Functions...
		Functions: inverse functions and their derivatives, derivatives of trig and inverse trig functions, exponential and logarithmic functions and their derivatives...
	functions.faasv.com /index.php?k=derivatives-of-trig-functions (648 words)

	Inverse functions and differentiation - Definition up Erdmond.Com (Site not responding. Last check: 2007-09-17)
		of a function y = f(x) is a function that, in some fashion, "undoes" the effect of f (see inverse_function for a formal and detailed definition).
		\frac{dy}{dx} denotes the derivative of the function y=f(x) with respect to x.
		\frac{dx}{dy} denotes the derivative of the function x=f(y) with respect to y.
	www.erdmond.com /Inverse_functions_and_differentiation.html (342 words)

	Inverse functions and differentiation
		The inverse of a function $y = f(x)$ is a function that, in some fashion, "undoes" the effect of $f$ (see inverse function for a formal and detailed definition).
		$y = e^x$ has inverse $x = \ln (y)$ (for positive $y$ ).
		The text of this article is licensed under the GFDL.
	www.ebroadcast.com.au /lookup/encyclopedia/in/Inverse_functions_and_differentiation.html (239 words)

	djb:Calculus I
		This course is intended to develop an understanding of the basic concepts of plane analytic geometry, the limit process, continuity of functions, differentiation, integration, and areas under and between curves.
		These included the use of first and second derivatives as aids in graphing functions, extrema problems, approximation of areas of plane regions, and selected applications of the calculus in the areas of business, economics and physical sciences.
		Use differentiation rules to obtain derivatives of algebraic, trigonometric, logarithmic, exponential, and composite functions.
	www.math-cs.cmsu.edu /~dbachman/math1151 (343 words)

	Edinburgh Mathematics Programme
		A substantial component is devoted to the solution of differential equations, perhaps the single most useful application of calculus in the physical sciences and engineering.
		Differentiation of inverse functions and its use in integration.
		Differentials, with applications to error estimation, Newton-Raphson's method and differentiation of implicitly defined functions.
	www.maths.ed.ac.uk /~derek/Syll/mm2.html (508 words)

	Hyperbolic Trigonometric Functions
		We define the Arcsinh(x) function as the principal inverse of the hyperbolic sine function.
		This is a first-order ordinary differential-equation in the radius r = 1 / p as a function of the angle theta.
		The values of the inverse hyperbolic trigonometric functions have to be obtained from the foregoing arctangent, by solving the quadratic equations of the identities and
	www.rism.com /Trig/hyperbol.htm (9675 words)

	Derivative
		In finance, derivative is the common short form for derivative security.
		Jerk is the derivative (with respect to time) of an object's acceleration.
		for all functions f and g and all real numbers a and b.
	www.sciencedaily.com /encyclopedia/derivative (1888 words)

	4.1.2.2 Differentiation Of The Inverse Trigonometric Functions
		Differentiate each of the following functions, simplifying the answer when appropriate.
		Section 3.1 Theorem 6.1 that if a function f is continuous on [a, b] and its derivative is 0 on
		Note that we didn't “boast” about the derivatives of the inverse cofunctions arccos x, arccot x, and arccsc x.
	www.geocities.com /pkving4math2tor4/4_the_elem_transc_func/4_01_02_02_diffn_of_the_inv_trig_func.htm (337 words)

	Inverse Trigonometric Functions and Differentiation (Site not responding. Last check: 2007-09-17)
		WHY?: This is because all of the trigonometric functions are periodic.
		This is the process of deriving an arc trig function (sine).
		When finding the derivative of an arctrig function, first change the function into a normal trig function of y = x.
	www.kent.k12.wa.us /staff/DavidWright/calculus/book/58 (171 words)

	CALCULUS homework problems (Site not responding. Last check: 2007-09-17)
		Determine whether f(x), the original function, is increasing (when f '(x) >0) or decreasing (when f '(x) <0) on each interval.
		The critical value for which f(x) is increasing to the left and decreasing to the right is a relative max.
		Sketch the function f and indicate the region on the interval [a,b].
	www.battaly.com /calc/calchw.htm (1254 words)

	inverse differentiation calculus (Site not responding. Last check: 2007-09-17)
		The inverse of f is denoted f - 1.
		Tutorial on using the derivative to detect increasing and decreasing functions.
		The inverse trigonometric functions have their own rules for differentiation.
	learning-gd.com /articles/315/inverse-differentiation-calculus.html (126 words)

	Exambot - Differentiation Quiz 2 (trig and inverse trig functions) (Site not responding. Last check: 2007-09-17)
		Exambot - Differentiation Quiz 2 (trig and inverse trig functions)
		To properly test yourself, don't peek at the answers, and click on the "Mark my exam" button at the bottom when you are done.
		Differentiation Quiz 2 (trig and inverse trig functions)
	www.exambot.com /cgi/exam/show.cgi/math/difc/ctra/2005.ex?a (85 words)

	World Web Math: Inverse Functions (Site not responding. Last check: 2007-09-17)
		As with the chain rule, the Leibnitz notation can often provide insight that can be confirmed using the definition of the derivative or previously proven results.
		It may seem that the formalism doesn't add much to our toolkit for differentiation, and indeed, the above examples do not make a great case for memorizing the rule for differentation of inverse functions.
		However, when more complicated functions are considered, the use of the chain rule and implicit differentiation allows derivatives to be calculated with relative ease.
	web.mit.edu /wwmath/calculus/differentiation/inverse.html (200 words)

	MA1101 Introduction to Mathematics 2005/6 (Site not responding. Last check: 2007-09-17)
		Geometry: Cartesian plane, plane curves and graphs of elementary functions (linear, quadratic, hyperbola, circle, ellipse, exp, ln functions).Trigonometric functions, relation to right triangle geometry, graphs of trigonometric functions, sum of angle formulas, sine and cosine rule.
		Intermediate Value Theorem, Rolle's Theorem, Mean Value Theorem and application to zeros of functions and theorems relating derivatives with maximum-minimum, increasing-decreasing, concavity, constant functions and others.
		Application of derivatives and limits to curve tracing and function description (decreasing, increasing, concavity, maxima and minima).
	www.tekcities.com /dwholtby/ma1101/outline.html (216 words)

	Inverse: Definition and Links by Encyclopedian.com - All about Inverse
		Inverse: Definition and Links by Encyclopedian.com - All about Inverse
		Inverse functions and differentiation - How to differentiate inverse functions.
		Inverse discrete cosine transform - Opposite of some transformation.
	www.encyclopedian.com /in/Inverse.html (212 words)

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