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Topic: Implicit differentiation

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	Implicit function Summary
		Implicit differentiation is a technique used in calculus to compute the derivative of functions even when an explicit formula for the function is unknown.
		In general, the implicit function theorem states that an equation g(x, y) = C(where g is a continuously differentiable function and C is any constant) locally defines y as an implicit function of x in a neighborhood of any point where the partial derivative of g with respect to y is nonzero.
		Sometimes standard explicit differentiation cannot be used, and in order to obtain the derivative, another method such as implicit differentiation must be employed.
	www.bookrags.com /Implicit_function (931 words)

	Derivative - Wikipedia, the free encyclopedia
		Implicit differentiation: If f(x,y) = 0 is an implicit function, we have: dy/dx = - (∂f / ∂x) / (∂f / ∂y).
		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 (2214 words)

	Calculus - Open Encyclopedia (Site not responding. Last check: 2007-11-03)
		Differential calculus is concerned with finding the instantaneous rate of change (or derivative) of a function's value, with respect to changes of the function's arguments.
		Another application of differential calculus is Newton's method, an algorithm to find zeroes of a function by approximating the function by its tangents.
		The fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations.
	open-encyclopedia.com /Calculus (908 words)

	Implicit Differentiation (Site not responding. Last check: 2007-11-03)
		The "implicit differentiation problem" below is slightly too complex to discuss...
		Calculus I (Math 2413) - Derivatives - Implicit Differentiation...
		Math 1215 with Maple --- Implicit Differentiation Lab...
	www.scienceoxygen.com /math/170.html (81 words)

	Calculus at opensource encyclopedia (Site not responding. Last check: 2007-11-03)
		Differential calculus is concerned with finding the instantaneous rate of change (or derivative) of a function's value, with respect to changes within the function's arguments.
		Abel seems to have been the first to consider in a general way the question as to what differential expressions can be integrated in a finite form by the aid of ordinary functions, an investigation extended by Liouville.
		Arbogast (1800) was the first, however, to separate the symbol of operation from that of quantity in a differential equation.
	wiki.tatet.com /Calculus.html (1768 words)

	Implicit Differentiation
		Implicit differentiation is nothing more than a special case of the well-known chain rule for derivatives.
		This second method illustrates the process of implicit differentiation.
		It is important to note that the derivative expression for explicit differentiation involves x only, while the derivative expression for implicit differentiation may involve BOTH x AND y.
	www.math.ucdavis.edu /~kouba/CalcOneDIRECTORY/implicitdiffdirectory/ImplicitDiff.html (734 words)

	Derivative
		The simplest notation for differentiation that is in current use is due to Lagrange and uses the prime, ′.
		Leibniz's notation is versatile in that it allows one to specify the variable for differentiation (in the denominator).
		Implicit differentiation: If is an implicit function, we have: dy/dx = - (∂f / ∂x) / (∂f / ∂y).
	www.brainyencyclopedia.com /encyclopedia/d/de/derivative.html (1839 words)

	implicit differentiation (Site not responding. Last check: 2007-11-03)
		is: a form of differentiation using the chain rule.
		is used: for differentiating implicit functions which are defined by an equation that relates the dependent variable
		is done: by differentiating both sides of the equation, which yields, in general, an expression in
	www.pha.jhu.edu /~ggaspar/physics/glossary/glossary/ii/impldiff.htm (52 words)

	Derivative - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-11-03)
		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.
		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.
		For differentiation of complex functions of a complex variable see also Holomorphic function.
	encyclopedia.learnthis.info /d/de/derivative.html (1805 words)

	Implicit derivative - preview (Site not responding. Last check: 2007-11-03)
		Implicit derivatives come into play whenever a curve, a surface or, in general, a manifold is characterized as the solution set of one or several examples.
		Differentiating this equation with respect to x is called "implicit differentiation with respect to x".
		Differentiating this equation with respect to y is called "implicit differentiation with respect to y".
	www.ualberta.ca /dept/math/gauss/fcm/calculus/multvrbl/basic/ImplctFnctns/implct_drvtvs1.htm (180 words)

	Implicit Differentiation
		This is the differentiation of an equation which implicitly defines a function y of x.
		Observe that we differentiate both sides of the equation, and that since y is a function of x, the chain rule must be used for every y-term.
		When we implicitly differentiate an x-y equation, we tacitly assume that whatever function implicitly defined by the equation is differentiable.
	instruct.tri-c.edu /ilevina/docs/1620_files/implicit_differentiation.htm (1137 words)

	Implicit Differentiation - Science Articles
		Implicit Differentiation is a technique used to differentiate a function when it cannot be explicitly solved for the dependant variable.
		The basic method for implicit differentiation involves differentiating both sides of the equation with respect to x while treating y as being implicitly defined as a function of x.
		First differentiate both sides of the function with respect to x or whatever the independent variable may be.
	www.physicspost.com /articles.php?articleId=18 (469 words)

	Implicit Differentiation
		Implicit differentiation is a process which will clarify this for us.
		In fact, we can use implicit differentiation to verify the power rule for any fractional exponenent.
		If we compute the derivative at a point through implicit differentiation and it is defined at this point, then the Implicit Function Theorem justifies the calculation we have just done and we need not worry about anything.
	www.ugrad.math.ubc.ca /coursedoc/math100/notes/derivative/implicit.htmx (952 words)

	Qrhetoric Calculus - Implicit and Logarithmic Differentiation
		The only way to solve it is to utilize implicit differentiation, in a way.
		This generally ends up working slightly different from regular implicit differentiation, but it is very intuitive.
		We don’t do that by implicit differentiation, but here it is simple to, because we just plug in from the original equation.
	www.qcalculus.com /cal05.htm (960 words)

	Fundamental theorem of calculus (Site not responding. Last check: 2007-11-03)
		The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverses of each other.
		This means that if a continuous function is first integrated and then differentiated, the original function is retrieved.
		An important consequence of this, sometimes called the second fundamental theorem of calculus, allows one to compute integrals by using an antiderivative of the function to be integrated.
	www.sciencedaily.com /encyclopedia/fundamental_theorem_of_calculus_1 (571 words)

	Implicit Differentiation
		Implicit Differentiation examples with detailed solutions are presented
		Use the sum rule of differentiation to the whole term on the left of the given equation.
		Use the differentiation of a sum formula to left side of the given equation.
	www.analyzemath.com /calculus/Differentiation/implicit.html (372 words)

	Differentiation of Functions of Several Variables
		We conclude with two chapters which are really left over from last year's calculus course, and which should help to remind you of the techniques you met then.
		We shall mainly be concerned with differentiation and integration of functions of more than one variable.
		In this chapter we concentrate on differentiation, and in the last one, move on to integration.
	www.maths.abdn.ac.uk /~igc/tch/ma2001/notes/node63.html (69 words)

	Honors Elem. Calculus II Lecture Notes, 01/18/01 (Site not responding. Last check: 2007-11-03)
		The technique of implicit differentiation is a method for finding dy/dx when given an equation that defines y implicitly as a function of x -- as opposed to an equation of the form y = f(x), that defines y explicitly in terms of x
		One important application of implicit differentation is to generate derivatives of commonly used inverse functions, such as the inverse trigonometric functions.
		The differentiation formulas for the inverse hyperbolic sine, tangent, and secant are somewhat similar to their counterparts in the ordinary inverse trigonometric functions.
	www.assumption.edu /alfano/MAT132-SP01/Notes/011801.html (580 words)

	Steps for Implicit Differentiation (Site not responding. Last check: 2007-11-03)
		using implicit differentiation, we make some assumptions that are satisfied by most implicitly defined functions at most points.
		Whenever you differentiate and expression in x, differentiate as you did in previous work.
		Whenever you differentiate an expression in y, think of y as a function of x and use the chain rule.
	www.gpc.edu /~jcraig/cal1_ch3/3s6_implicit_diferentiation.htm (294 words)

	SparkNotes: Computing Derivatives: Techniques of Differentiation
		But we still may want to know the slope of the graph at a particular point, that is, the derivative of an implicit function at that point.
		The idea is to differentiate both sides of the equation with respect to x (using the chain rule where necessary).
		We can put the chain rule and implicit differentiation to work to find the derivative of an inverse function when we already know the derivative of the function itself.
	www.sparknotes.com /math/calcbc1/computingderivatives/section2.rhtml (773 words)

	Implicit Differentiation
		So far, all the functions being differentiated are explicit functions, meaning that one of the variables was specifically given in terms of the other variable.
		However, not all functions are given explicitly and are only implied by an equation.
		Implicit differentiation is taking the derivative of both sides of the equation with respect to one of the variables.
	library.thinkquest.org /10030/3implic.htm (186 words)

	Qrhetoric Calculus - Implicit and Logarithmic Differentiation
		The only way to solve it is to utilize implicit differentiation, in a way.
		This generally ends up working slightly different from regular implicit differentiation, but it is very intuitive.
		We don’t do that by implicit differentiation, but here it is simple to, because we just plug in from the original equation.
	calculus.freehomepage.com /cal05.htm (748 words)

	Math 1215 with Maple --- Implicit Differentiation Lab
		Differentiation in Maple does not always work quite the way you might expect.
		Differentiate the equation xy^3 + x^3y = 30 with respect to x, keeping the chain rule in mind.
		Work through the implicit differentiation example given in the examples and discussion section.
	www.math.utah.edu /~johnson/calcA98/implicit.html (1278 words)

	Calculus I Notes, Section 3-6
		Because y is not native to what are differentiating with respect to, we need to regard it as a composite function.
		Because we are differentiating with respect to x, we need to use the chain rule on the y.
		Because we are differentiating with respect to x, we need to use the chain rule on the left side.
	www.blc.edu /fac/rbuelow/calc/nt3-6.html (564 words)

	Solving differential equations (Site not responding. Last check: 2007-11-03)
		The simplest notation for differentiation that is in current use is due to
		Leibniz's notation is versatile in that it allows one to specify the variable for differentiation (in the denominator).
		Newton's notation for differentiation was to place a dot over the function name:
	math-tables.net /note.html (259 words)

	Implicit Differentiation
		An implicit function is one that cannot be easily manipulated to the form y = f(x).
		To find dy/dx, every term in the equation is differentiated with respect to x.
		We shall use this to illustrate the idea of differentiation of implicit functions.
	library.thinkquest.org /C0110248/calculus/difnimpt.htm (114 words)

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