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Topic: Chain rule

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	PlanetMath: chain rule
		The chain rule has a particularly suggestive appearance in terms of the Leibniz formalism.
		Rather, the Leibniz format is well suited to the interpretation of the chain rule in terms of related rates.
		This is version 9 of chain rule, born on 2002-02-24, modified 2004-09-27.
	planetmath.org /encyclopedia/ChainRule.html (108 words)

	Chain rule
		The Chain Rule is a formula for the derivative of the composition of two functions.
		The General Power Rule (GPR) is derivable, via the Chain Rule.
		The chain rule is a fundamental property of all definitions of derivative and is therefore valid in much more general contexts.
	www.ebroadcast.com.au /lookup/encyclopedia/ch/Chain_rule.html (365 words)

	The chain rule* (Site not responding. Last check: 2007-11-06)
		The chain rule is the rule we use if we want to take the derivative of a composition of functions.
		The one-variable chain rule states that the derivative of h is the product of the derivative of f and the derivative of g.
		Equation (5) shows that the chain rule in our two-variable case is just like the one-variable chain rule (equation (1)) applied twice.
	www.math.umn.edu /~nykamp/m2374/readings/chainrule (850 words)

	Visual 2 Variable Chain Rule
		[Recall that the single variable rule is that (fg)'(x)=f'(g(x)*g'(x).] Since slopes are heights of triangles, we multiply by using similar triangles, one triangle having a base of 1 and a height of g'(x), the other having a base of f'(g(x)).
		The third step is to notice that the rule works as we move through a variety of values of t0.
		Using the readouts on the control panel, the chain rule formula works as the value of t0 moves through a range.
	www.slu.edu /classes/maymk/banchoff/ChainRule2D.html (528 words)

	Karl's Calculus Tutor - 4.4 Derivatives: Chain Rule Applications
		The chain rule is admittedly the most difficult of the rules we have encountered so far.
		This is because the chain rule's usefulness goes beyond the problem of finding the derivative of something that is explicitly the composite of two functions.
		So before proceeding with this section, be sure that you understand the statement of the chain rule and the example that follows it.
	www.karlscalculus.org /calc4_4.html (3017 words)

	The Chain Rule
		As a motivation for the chain rule, consider the function
		The answer is given by the Chain Rule.
		Then the Chain rule implies that f'(x) exists, which we knew since it is a polynomial function, and
	www.sosmath.com /calculus/diff/der04/der04.html (312 words)

	Calculus III (Math 2415) - Partial Derivatives - Chain Rule (Site not responding. Last check: 2007-11-06)
		We’ve been using the standard chain rule for functions of one variable throughout the last couple of sections. It’s now time to extend the chain rule out to more complicated situations. Before we actually do that let’s first review the notation for the chain rule for functions of one variable.
		Here is the chain rule for both of these cases.
		Okay, now that we’ve seen a couple of cases for the chain rule let’s see the general version of the chain rule.
	tutorial.math.lamar.edu /AllBrowsers/2415/ChainRule.asp (1252 words)

	Chain rule (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-11-06)
		In calculus, the chain rule is a formula for the derivative of the composition of two functions.
		In algebraic terms, the chain rule (of one variable) states that if functions f and g are both differentiable and function F is defined as f composed with g, that is :
		As an advanced explanation of the tensor concept, one can interpret the chain rule as applied to coordinate changes also as the requirement for self-consistent concepts of tensor giving rise to tensor fields.
	chain-rule.kiwiki.homeip.net.cob-web.org:8888 (743 words)

	Chain rule - Wikipedia, the free encyclopedia
		In integration, the counterpart to the chain rule is the substitution rule.
		See tensor field for an advanced explanation of the fundamental role the chain rule plays in the geometric nature of tensors.
		Faà di Bruno's formula generalizes the chain rule to higher derivatives.
	en.wikipedia.org /wiki/Chain_rule (438 words)

	3 (Site not responding. Last check: 2007-11-06)
		to apply the chain rule, we need to be able to express a complicated function as a composition of simpler functions (an inside function and an outside function) whose derivatives we know.
		In words, the chain rule says that the derivative of the composition of two differentiable functions is the derivative of the outside function evaluated at the inside function times the derivative of the inside function.
		One useful special case of the chain rule is the generalized power rule.
	www.gpc.edu /~jcraig/cal1_ch3/3s5_chain_rule.htm (267 words)

	Applying the Chain Rule
		I've been having some trouble grasping the conditions necessary to apply the chain rule to achieve the derivative of an algebraic expression or even apply it to a real world situation.
		Essentially, I believe the Chain Rule is applied when it is possible to seperate a function into two seperate algebraic equations.
		Chain Rule works indeed for what seems to be a composite function or something that is made out of more components.
	www.physicsforums.com /showthread.php?t=97556 (481 words)

	The Multivariable Chain Rule - HMC Calculus Tutorial
		Although the formal proof is not trivial, the variable-dependence diagram shown here provides a simple way to remember this Chain Rule.
		Again, the variable-dependence diagram shown here indicates this Chain Rule by summing paths for z either to u or to v.
		These Chain Rules generalize to functions of three or more variables in a straight forward manner.
	www.math.hmc.edu /calculus/tutorials/multichainrule (279 words)

	Math Forum - Ask Dr. Math Archives: Chain Rule
		I have a question on the "chain rule" when finding the derivatives of polynomials.
		I think I need to use l'Hopital's Rule and the natural logarithm function to find lim (x goes to 0) [(cos (2x))^(3/(x^2))].
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /library/drmath/sets/select/dm_chain_rule.html (195 words)

	World Web Math: The Chain Rule (Site not responding. Last check: 2007-11-06)
		This rule is called the chain rule because we use it to take derivatives of composties of functions by chaining together their derivatives.
		The chain rule can be thought of as taking the derivative of the outer function (applied to the inner function) and multiplying it times the derivative of the inner function.
		It is commonly where most students tend to make mistakes, by forgetting to apply the chain rule when it needs to be applied, or by applying it improperly.
	web.mit.edu /wwmath/calculus/differentiation/chain.html (352 words)

	Visual Calculus - Chain Rule (Site not responding. Last check: 2007-11-06)
		Objectives: In this tutorial, we derive the Chain Rule.
		After working through these materials, the student should be able to use the Chain Rule to differentiate certain functions.
		LiveMath notebook which illustrates the use of the chain rule.
	archives.math.utk.edu /visual.calculus/2/chain_rule.4/index.html (62 words)

	Chain Rule
		The chain rule is a rule for differentiating compositions of functions.
		However, we rarely use this formal approach when applying the chain rule to specific problems.
		Function f is the ``outer layer'' and function g is the ``inner layer.'' Thus, the chain rule tells us to first differentiate the outer layer, leaving the inner layer unchanged (the term f'(g(x))), then differentiate the inner layer (the term g'(x)).
	www.math.ucdavis.edu /~kouba/CalcOneDIRECTORY/chainruledirectory/ChainRule.html (455 words)

	The chain rule - An approach to calculus
		The chain rule can be extended to more than two functions.
		Assume that y is a function of x, and apply the chain rule to express each derivative with respect to x.
		To prove the chain rule let us go back to basics.
	www.themathpage.com /acalc/chain.htm (468 words)

	Applications of The Chain Rule
		The chain rule can be applied to determining how the change in one quantity will lead to changes in the other quantities related to it.
		Applying the chain rule to calculate the derivative shown in this expression, we have
		(b) A related application of the chain rule to a chemical situation concerns the relationship between a measure of the average speed of molecules in a gas and the temperature.
	www.ugrad.math.ubc.ca /coursedoc/math100/notes/derivative/chainap.html (1189 words)

	Chain Rule (Site not responding. Last check: 2007-11-06)
		The purpose of this applet is to illustrate the chain rule:
		Tangent lines are drawn at these points, as well as the slope of each line, at each of these points, in order to illustrate the chain rule.
		The user may enter functions of this form in the "f(x)=" and "g(x)=" input box or use the mouse to define new functions.
	www.scottsarra.org /applets/calculus/FunctionComposition.html (208 words)

	Chain Rule -- from Wolfram MathWorld (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-11-06)
		There are a number of related results that also go under the name of "chain rules." For example, if
		The "general" chain rule applies to two sets of functions
		Kaplan, W. "Derivatives and Differentials of Composite Functions" and "The General Chain Rule." §2.8 and 2.9 in
	mathworld.wolfram.com.cob-web.org:8888 /ChainRule.html (109 words)

	Chain Rule for Partial Derivatives (Site not responding. Last check: 2007-11-06)
		In the process we will explore the Chain Rule applied to functions of many variables.
		A function is a rule that assigns a single value to every point in space, e.g.
		Essentially the same procedures work for the multi-variate version of the Chain Rule.
	www.math.usu.edu /~powell/math320/node5.html (286 words)

	The Chain Rule
		Before we write down the Chain Rule, let's think about our earlier example.
		This last question is important and sometimes a source of confusion when understanding the general Chain Rule.
		As becomes small, the approximation improves so that in the limit, we find the chain rule.
	www.ugrad.math.ubc.ca /coursedoc/math100/notes/derivative/chain.html (290 words)

	Introductory Calculus: The Chain Rule (Site not responding. Last check: 2007-11-06)
		Note that in this lesson we show examples, but not any proof that the formulas for the product and quotient rules are correct.
		There is another approach to this derivative which helps us with more complicated functions.
		Write in fraction form, if needed, so that all exponents are positive in your final answer.
	www.algebralab.org /studyaids/studyaid.aspx?file=Calculus_6-29B.xml (123 words)

	The Chain Rule (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-11-06)
		The chain rule (function of a function) is very important in differential calculus and states that:
		This rule allows us to differentiate a vast range of functions.
		by the Chain Rule, dy/dx = dy/dt × dt/dx
	www.mathsrevision.net.cob-web.org:8888 /alevel/pages.php?page=42 (146 words)

	The Chain Rule - HMC Calculus Tutorial (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-11-06)
		The Chain Rule - HMC Calculus Tutorial (via CobWeb/3.1 planetlab2.netlab.uky.edu)
		The three formulations of the Chain Rule given here are identical in meaning.
		Sometimes you will need to apply the Chain Rule several times in order to differentiate a function.
	www.math.hmc.edu.cob-web.org:8888 /calculus/tutorials/chainrule (116 words)

	The Chain Rule (Site not responding. Last check: 2007-11-06)
		The chain rule lets us figure out the derivative of the composition of two functions.
		When we're thinking abstractly, compositions are things that are written like:
		In both cases, this is the function that says "Take x, first compute the function g with input x, next, apply the function f to the result of the first operation."
	www.math.uic.edu /~fields/calc_tutorial/chain_rule (57 words)

	The CRM114 Discriminator - The Controllable Regex Mutilator
		CRM114 is a system to examine incoming e-mail, system log streams, data files or other data streams, and to sort, filter, or alter the incoming files or data streams according to the user's wildest desires.
		Criteria for categorization of data can be by satisfaction of regexes, by sparse binary polynomial matching with a Bayesian Chain Rule evaluator, a Hidden Markov Model, or by other means.
		Accuracy of the SBPH/BCR classifier has been seen in excess of 99 per cent, for 1/4 megabyte of learning text.
	crm114.sourceforge.net (2332 words)

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