21. Karl's Calculus Tutor - 4.4 Derivatives: Chain Rule Applications Click here to go right to Implicit differentiation. 4.4 Chain Rule Applications. Throughoutcalculus we will be making substitutions of variables. http://www.karlscalculus.org/calc4_4.html

Section 4: Derivatives Click here to go right to Implicit Differentiation 4.4 Chain Rule Applications The chain rule is admittedly the most difficult of the rules we have encountered so far. But it is also the most powerful. You must be able to apply the mechanics of this rule before you will be ready for the next challenge, which is knowing when to apply it. 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. Throughout calculus we will be making substitutions of variables. It happens all the time. And every time we do, the chain rule will apply. And there are other applications, as we shall see shortly. So before proceding with this section, be sure that you understand the statement of the chain rule and the example that follows it. Review it until you have some confidence in your grasp of it. If reviewing the story about the professor's watch helps, then review that as well. Then come back here and see if you can do the exercises that follow. When you can do the exercises, then procede to what follows them. Exercises Apply the chain rule to find the derivatives of the following. Check your work by expanding the expression shown below and using other methods we have learned to arrive at the same answer.

22. PinkMonkey.com Calculus Study Guide - Section 4.13 Logarithmic Differentiation 4.13 Logarithmic differentiation. In order to find the derivative of The processis known as the logarithmic differentiation. Example 61 If y = x x Find y’. http://www.pinkmonkey.com/studyguides/subjects/calc/chap4/c0404d01.asp

Support the Monkey! Tell All your Friends and Teachers Get Cash for Giving Your Opinions! Free College Cash 4U Advertise Here See What's New on the Message Boards today! Favorite Link of the Week. Your house from the sky! ... Tell a Friend Win a $1000 or more Scholarship to college! Place your Banner Ads or Text Links on PinkMonkey for $0.50 CPM or less! Pay by credit card. Same day setup. Please Take our User Survey or g ( x ) where f ( x ) and g ( x ) are both derivable, it is usually advisable to take logarithm of the function first and then differentiate. The process is known as the logarithmic differentiation. Example 61 x Solution : x x Differentiating w. r. to x, we get Example 62 log x Solution: log x then we have log x next page Index 4. 1 Derivability At A Point 4. 2 Derivability In An Interval ... 4.11 Derivatives Of Inverse Trigonometric Functions 4.13 Logarithmic Differentiation 4.14 Derivatives Of Functions In Parametric Form 4.15 Higher order Derivatives Chapter 5 4659 PinkMonkey users are on the site and studying right now.

23. PinkMonkey.com Calculus Study Guide - Section 4.9 Differentiation Of Implicit Fu 4.9 differentiation of Implicit Functions. If an wr.to. y and thenmultiply it by to get its differentiation wr.to x. next page. http://www.pinkmonkey.com/studyguides/subjects/calc/chap4/c0404901.asp

Support the Monkey! Tell All your Friends and Teachers Get Cash for Giving Your Opinions! Free College Cash 4U Advertise Here See What's New on the Message Boards today! Favorite Link of the Week. Your house from the sky! ... Tell a Friend Win a $1000 or more Scholarship to college! Place your Banner Ads or Text Links on PinkMonkey for $0.50 CPM or less! Pay by credit card. Same day setup. Please Take our User Survey If an equation is expressed as y = f(x), then y is said to be the explicit function of x. However if y is connected with x by an expression f (x, y) = 0, then y is said to an implicit function of x. For e.g. x + 3xy + y For differentiating the implicit function we differentiate each term w.r.to x keeping in mind that if there is a term contains powers of y, we differentiate first w.r.to. y and then multiply it by to get its differentiation w.r.to x. next page Index 4. 1 Derivability At A Point 4. 2 Derivability In An Interval ... 4. 8 Derivative Of A Composite Function 4. 9 Differentiation Of Implicit Functions 4.10 Derivative Of An Inverse Function

24. World Web Math: Calculus Index calculus Index. Paths Independent Study. Topics Summary; differentiation Tableof derivatives; Definition of differentiation; The Notation of differentiation; http://web.mit.edu/wwmath/calculus/

Calculus Index Paths: Independent Study Topics: Summary Limits Formal definition of limits The Squeeze Theorem Useful Trigonometric Limits Differentiation Table of derivatives Definition of differentiation The Notation of Differentiation Rules of Differentiation Derivatives of polynomial functions The Product Rule The Quotient Rule The chain rule ... Differentiation of inverse functions Derivatives of Trigonometric functions Derivative of sine Derivatives of cosine Derivatives of other trigonometric functions Derivatives of inverse trigonometric functions Derivatives of Logarithmic Functions Euler's magic number/ Derivative of exponentials Differentiation of logarithms Derivatives of hyperbolic functions Applications of Differentiation Integration Methods of integration Back to the World Web Math Home Page Back to the World Web Math Categories Page watko@mit.edu Last modified November 5, 1998

25. Index Of /wwmath/calculus/differentiation Parent Directory 05Jun-2000 1614 - chain-proof.html......Index of /wwmath/calculus/differentiation. Name Last modified Size http://web.mit.edu/wwmath/calculus/differentiation/

Index of /wwmath/calculus/differentiation Name Last modified Size Description ... Parent Directory 05-Jun-2000 16:14 - chain-proof.html 26-Aug-1998 16:04 2k chain-solutions.html 26-Aug-1998 12:57 2k chain.html 03-Oct-1999 16:59 4k chaineq/ 08-Apr-2002 20:12 - cos-solutions.html3 24-Aug-1998 12:10 1k cos.html3 24-Aug-1998 12:10 2k defeq/ 14-Oct-1999 17:59 - definition.html 14-Oct-1999 17:52 4k e-rules.html3 24-Aug-1998 12:10 2k eq.log 29-Aug-1998 16:53 0k euler-solutions.html3 24-Aug-1998 12:10 2k euler.html3 24-Aug-1998 12:10 7k figs/ 02-Sep-1998 17:00 - fraceq/ 07-Sep-1998 14:37 - fract-solutions.html 07-Sep-1998 14:51 1k fractional.html 07-Sep-1998 14:41 3k hyper-solutions.html3 24-Aug-1998 12:10 1k hyperbolic.html 06-Nov-2001 13:17 4k hyperbolic.html3 24-Aug-1998 12:10 4k impl-solutions.html

26. Partial Differentiation - HMC Calculus Tutorial Partial differentiation Dx. =, lim Dx®0, 2xy + xDx. =, 2xy, In practice, weuse our knowledge of singlevariable calculus to compute partial derivatives. http://www.math.hmc.edu/calculus/tutorials/partialdifferentiation/

Partial Differentiation Suppose you want to forecast the weather this weekend in Los Angeles. You construct a formula for the temperature as a function of several environmental variables, each of which is not entirely predictable. Now you would like to see how your weather forecast would change as one particular environmental factor changes, holding all the other factors constant. To do this investigation, you would use the concept of a partial derivative Let the temperature T depend on variables x and y, T = f(x,y). The rate of change of f with respect to x (holding y constant) is called the partial derivative of f with respect to x and is denoted by f x (x,y). Similarly, the rate of change of f with respect to y is called the partial derivative of f with respect to y and is denoted by f y (x,y). We define f x (x,y) = lim D x f(x+ D x,y)-f(x,y) D x f y (x,y) = lim D x f(x,y+ D x)-f(x,y) D x Do you see the similarity beween these and the limit definition of a function of one variable? Example Let f(x,y) xy Then f x (x,y)

27. Qrhetoric Calculus - Implicit And Logarithmic Differentiation Advanced differentiation. Implicit differentiation. This will only be easy if youcan follow the steps. Logarithmic differentiation. OK, here’s the deal. http://calculus.freehomepage.com/cal05.htm

Study Sheets Advanced Differentiation Problem Sets (Non Javascript - Problems Answers Implicit Differentiation This will only be easy if you can follow the steps. Understanding it is nice too, but why waste your time with something so trivial as actually knowing what it is you are learning. In this section, you are taking a derivative, except the issue raised here is that you cannot always isolate y onto one side of your equation, making a different method necessary. Here is a typical example: = 2x + x y Actually, this one’s a bit tough, I think. Anyway, you should already see that it is impossible to take a regular derivative. y is not a simple thing off to one side, and to isolate y off to one side would be difficult. So we are going to take a straight derivative of each or the 3 terms here, and then try to isolate y’, which is much easier. That way we will have a derivative for the equation. One important thing to remember when doing implicit differentiation: The derivative of y is y’. The derivative of x is taken in the normal way of a variable, so that the derivative of x is 1, as an example. The reason for this is that y is a function of x, and does not exist without being dependant on the equation on the other side, on the other hand, x exists all along the domain. I mentioned this briefly in my overview of derivatives.

28. The Math Forum - Math Library - Calculus (SV) deltaepisilon proof, definition of a derivative, differentiation rules, trig derivatives,the chain rule, implicit differentiation; calculus II definition of http://mathforum.org/library/topics/svcalc/

Browse and Search the Library Home Math Topics : Calculus (SV) Library Home Search Full Table of Contents Suggest a Link ... Library Help Subcategories (see also All Sites in this category Differentiation Integration Sequences and Series Selected Sites (see also All Sites in this category Calculus and Differential Equations (MathPages) - Kevin Brown About 40 "informal notes" by Kevin Brown on calculus and differential equations: limit paradox, proofs that pi and e are irrational, Ptolemy's Orbit, leaning ladders, how Leibniz might have anticipated Euler, and many more. more>> CALCULUS@INTERNET - WebPrimitives, Cambridge, Massachusetts Calculus on the Web. Contents include: PreCalculus (functions, algebra, trigonometry); Calculus Topics (differential and integral calculus, sequences and series, multivariable calculus, differential equations); Assistance (student, instructor); Curriculum Material; Technology (graphing calculators, CAS); Assessment; Courses; Reference Material; Applications; Pedagogy; Advanced Mathematics (analysis, algebra, number theory); links for Kids and Recreation. more>> Calculus - Math Forum Links to some of the best Internet resources for calculus: classroom materials, software, Internet projects, and public forums for discussion.

29. [ap-calculus] RE: Symbolic Differentiation Utility? a topic from APCALC ap-calculus RE symbolic differentiation utility?post a message on this topic post a message on a new topic http://mathforum.org/epigone/ap-calc/drahslarkor

a topic from AP-CALC [ap-calculus] RE: symbolic differentiation utility? post a message on this topic post a message on a new topic 7 Oct 2002 [ap-calculus] RE: symbolic differentiation utility? , by James Eggleston 8 Oct 2002 [ap-calculus] Re: symbolic differentiation utility? , by Cdmath5221@aol.com 8 Oct 2002 [ap-calculus] Re: symbolic differentiation utility? , by John Stanley 8 Oct 2002 [ap-calculus] Re: symbolic differentiation utility? , by Janet Robert 8 Oct 2002 [ap-calculus] Re: symbolic differentiation utility? , by M J Bell 8 Oct 2002 [ap-calculus] Re: symbolic differentiation utility? , by Jonathan Eli Wickham 9 Oct 2002 [ap-calculus] Re: symbolic differentiation utility? , by LnMcmullin@aol.com 9 Oct 2002 [ap-calculus] Re: symbolic differentiation utility? , by Ed Isaacs 9 Oct 2002 [ap-calculus] Re: symbolic differentiation utility? , by Ruth Levenstein 10 Oct 2002 [ap-calculus] Re: symbolic differentiation utility? , by James Eggleston 14 Oct 2002 [ap-calculus] RE: symbolic differentiation utility? , by JVB 14 Oct 2002 [ap-calculus] Re: symbolic differentiation utility?

30. Calculus/Differentiation next up previous Next Maxima/Minima Up Math Review Previous TrigonometryCalculus/differentiation. 1. if p = f(T) then. Thus, if p = T 2 then. http://www.sp.uconn.edu/~cdavid/mathrev2/node2.html

Next: Maxima/Minima Up: Math Review Previous: Trigonometry Calculus/Differentiation if p = f(T) then Thus, if p T then From this it follows that (a) (b) and (c) and if h(t) and g(t) are functions of t, then if i(t) and j(t) are functions of t, then if k = n(m) and m is a function of of g, i.e., m = m(g), then This is an example of the chain rule. if y = f(x) has a solution x=g(y) then Maxima/Minima Series DeMoivre's (Euler's) Theorem etc

31. Calculus/Partial Differentiation Theorem etc calculus/Partial differentiation. 1. If f(x,y) is a functionof two variables, x and y, then. (a) If then. (b) If then. (c http://www.sp.uconn.edu/~cdavid/mathrev2/node6.html

Next: Calculus/Integration Up: Math Review Previous: DeMoivre's (Euler's) Theorem etc Calculus/Partial Differentiation If f(x,y) is a function of two variables, x and y, then (a) If then (b) If then (c) The double differentiation is order independent, i.e., If f(x,y) is a function of two variables, x and y, then If then If then The double differentiation is order independent, i.e.

32. ThinkQuest Library Of Entries This page is a reference sheet which contains key calculus differentiation and integrationformulas. Be sure to check the formula database for other formulas. http://library.thinkquest.org/20991/calc/reference.html

Welcome to the ThinkQuest Internet Challenge of Entries The web site you have requested, Math for Morons like Us , is one of over 4000 student created entries in our Library. Before using our Library, please be sure that you have read and agreed to our To learn more about ThinkQuest. You can browse other ThinkQuest Library Entries To proceed to Math for Morons like Us click here Back to the Previous Page The Site you have Requested ... Math for Morons like Us click here to view this site A ThinkQuest Internet Challenge 1998 Entry Click image for the Site Languages : Site Desciption Have you ever been stuck on math? If it was a question on algebra, geometry, or calculus, you might want to check out this site. It's all here from pre-algebra to calculus. You'll find tutorials, sample problems, and quizzes. There's even a question submittal section, if you're still stuck. A formula database gives quick access and explanations to all those tricky formulas. Languages: English. Students J. Robert Davis High School Library UT, United States

33. ThinkQuest Library Of Entries calculus Limits. The Derivatives. The Slope of a Curve differentiation andDerivatives Tangent and Normal Lines of a Curve Derivative Formulas. http://library.thinkquest.org/10030/calcucon.htm

Welcome to the ThinkQuest Internet Challenge of Entries The web site you have requested, Seeing is Believing , is one of over 4000 student created entries in our Library. Before using our Library, please be sure that you have read and agreed to our To learn more about ThinkQuest. You can browse other ThinkQuest Library Entries To proceed to Seeing is Believing click here Back to the Previous Page The Site you have Requested ... Seeing is Believing click here to view this site A ThinkQuest Internet Challenge 1997 Entry Click image for the Site Languages : Site Desciption Need a primer on math, science, technology, education, or art, or just looking for a new Internet search engine? This catch-all site covers them all. Maybe you're doing your homework and need to quickly look up a basic term? Here you'll find a brief yet concise reference source for all these topics. And if you're still not sure what's here, use the search feature to scan the entire site for your topic. Students Peter Oakhill College, Castle Hill Australia Suranthe H Oakhill College Australia Coaches Tina Oakhill College, Castle Hill

34. Math 583 B - Calculus Of Variations - Functional Differentiation Principles and Methods of Applied Mathematics calculus of Variations.Functional differentiation. If you find typos and/or have suggestions http://www.math.arizona.edu/~lega/583/Spring99/lectnotes/FD2.html

Principles and Methods of Applied Mathematics Calculus of Variations Functional Differentiation If you find typos and/or have suggestions regarding these notes, please send me an e-mail at lega@math.arizona.edu Back to MATH 583

35. Study Room - Mathematics - Calculus - Differentiation - Differentiation Of Ekx A Home, Study Room, Utilities, Links, Exam Hall, Forum, News Room, YourProfile, eStore, Logout. calculus I differentiation. Gradient of acurve. http://www.examstutor.com/maths/resources/studyroom/calculus_differentiation/dif

Home Study Room Utilities Links ... Logout Calculus I: differentiation Gradient of a curve Differentiation from first principles Differentiation of x n ... Applications Differentiation of e kx and ln x Differentiation of e kx and ln x Links Maths Utilities Graph Wizard Exam Hall Multiple Choice Differentiation of ekx and lnx Test Exam Papers AS, OCR, Spec 2000 A2, OCR, Spec 2000 A2, OCR, Spec 2000 A2, OCR, Spec 2000 Search Site advanced search... Differentiation of e kx and ln x If , then it can be shown that If , then it can be shown that If , then it can be shown that Example Find for each of the following: Biology Business Chemistry Mathematics ... Contact Details

36. Study Room - Mathematics - Calculus I Differentiation - Differentiation Of Xn An Study Room Mathematics - calculus I differentiation - differentiation of xn andAxn. Please enter your user name and password to gain access to this resource http://www.examstutor.com/maths/resources/studyroom/calculus_differentiation/dif

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37. MathMadeEasy.com 64 min. calculus Semester I Volume II differentiation Part I, Derivatives and tangentlines; velocity and other rates of change; differentiation rules for sums http://www.mathmadeeasy.com/calculussemesteri.html

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38. Calculus 1 Flipper Grades 11 College. Sample card from the calculus 1 Flipper. DerivativesBasicrules of differentiation; DerivativesComposition rule for differentiation; http://www.studyguidesinternational.com/188.htm

Calculus 1 Flipper Item #CLP -18 Cost $6.95 Contains over 375 rules, definitions, and examples. Grades 11 - College. Sample card from the Calculus 1 Flipper. Topics Covered Trigonometric identities Trigonometric formulas Length and line formulas Area formulas Volume and miscellaneous formulas LimitsInformal definition LimitsEvaluation of finite limits LimitsInfinite LimitsOne-sided LimitsL'Hopital's rule LimitsSpecial cases Continuity Continuity and differentiability DerivativeDefinitions Finding the derivative using the definition DerivativesBasic rules of differentiation DerivativesComposition rule for differentiation DifferentiationChain rule Differentiation rulesTrigonometric functions DifferentiationInverse trigonometric functions DifferentiationInverse trigonometric examples DifferentiationExponential functions DifferentiationLogarithmic functions DifferentiationImplicit differentiation DifferentiationHigher order derivatives DifferentiationParametric equations DifferentiationTangent and normal lines to a curve DifferentiationGraphical analysis DifferentiationFirst derivative test DifferentiationSecond derivative test DifferentiationMaximum/minimum problems DifferentiationRelated rate problems DifferentiationMean value theorem DifferentiationMiscellaneous theorems DifferentiationDistance, velocity, and acceleration

39. THE CALCULUS PAGE PROBLEMS LIST differentiation; Problems on the differential; Problems on the IntermediateValueTheorem; Problems on the Mean Value Theorem. Beginning Integral calculus http://www.math.ucdavis.edu/~kouba/ProblemsList.html

THE CALCULUS PAGE PROBLEMS LIST Problems and Solutions Developed by : D. A. Kouba And brought to you by : eCalculus.org Beginning Differential Calculus : Problems on the limit of a function as x approaches a fixed constant limit of a function as x approaches plus or minus infinity limit of a function using the precise epsilon/delta definition of limit limit of a function using L'Hopital's Rule Problems on the continuity of a function of one variable Problems on the "Squeeze Principle" Problems on the limit definition of the derivative Problems on the chain rule Problems on the product rule Problems on the quotient rule Problems on differentiation of trigonometric functions Problems on differentiation of inverse trigonometric functions Problems on detailed graphing using first and second derivatives Problems on applied maxima and minima Problems on implicit differentiation Problems on related rates Problems on logarithmic differentiation Problems on the differential Problems on the Intermediate-Value Theorem Problems on the Mean Value Theorem Beginning Integral Calculus : Problems using summation notation Problems on the limit definition of a definite integral Problems on u-substitution Problems on integrating exponential functions Problems on integrating trigonometric functions Problems on integration by parts Problems on integrating certain rational functions, resulting in

40. Differentiation METRIC differentiation. The differential calculus is the mathematical scienceof change. The ideas that underlie it began to be investigated http://metric.ma.ic.ac.uk/differentiation/

Differentiation The differential calculus is the mathematical science of change . The ideas that underlie it began to be investigated over two thousand years ago, but the great leap forward came with Newton and Leibniz in the seventeenth century. Their groundbreaking work made the mathematical study of change a practical proposition for the first time, but the logical foundations of calculus remained shaky until the nineteenth century, when the field was finally placed on a firm footing. Because the calculus is so important, it is sometimes talked about in hushed, reverent tones. This can give the impression that it's fearsomely difficult. But it isn't; actually, the whole reason for its importance is that it makes life easy for us. The central idea, as far as the modern (that is, nineteenth-century) version of calculus is concerned, is that of limits. Principles of Differentiation. Techniques of Differentiation. Applications of Differentiation.

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