The chain rule tells us how to find the derivative of a composite function. The chain rule is also useful in electromagnetic induction. This discussion will focus on the chain rule of differentiation. The book s aim is to use multivariable calculus to teach mathematics as a blend of reasoning, computing, and problemsolving, doing justice to the structure, the details, and the scope of the ideas. For example, the ideal gas law describes the relationship between pressure, volume, temperature, and number of moles, all of which can also depend on time. The problem that many students have trouble with is trying to figure out which parts of the function are within other functions i. You can start with an example such as finding the derivative of.
State the chain rule for the composition of two functions. Chapter 9 is on the chain rule which is the most important rule for di erentiation. Without this we wont be able to work some of the applications. Khan academy is a nonprofit with the mission of providing a free, worldclass education for anyone, anywhere.
This lesson contains the following essential knowledge ek concepts for the ap calculus course. Be able to guess an antiderivative of a function that has linear insides. Discussion of the chain rule for derivatives of functions. The chain rule, part 1 math 1 multivariate calculus d joyce, spring 2014 the chain rule. Calculus cheat sheet university of georgia pdf book.
Well start with the chain rule that you already know from ordinary functions of one variable. Implementing the chain rule is usually not difficult. Accompanying the pdf file of this book is a set of mathematica notebook files. In multivariable calculus, you will see bushier trees and more complicated forms of the chain rule where you add products of derivatives along paths. The derivative will be equal to the derivative of the outside function with respect to the inside, times the derivative of the inside function. Chain rule with tables get 3 of 4 questions to level up. To make the rule easier to handle, formulas obtained from combining the rule with simple di erentiation formulas are given. In addition, as the last example illustrated, the order in which they are done will vary as well. The outside function is the square root or the exponent of 1 2 depending on how you want to think of it and the inside function is the stuff that were taking the square root of or raising to the 1 2, again depending on how you want to look at it. Ill just take this moment to encourage you to work the problems in the videos below along with me, or even before you see how i do them, because the chain rule is definitely something where actually doing it is the only way to get better.
Lee lady for years, i used to tell people that i wished someone would write calculus for dummies, using the style of that popular series. Functions and their graphs, trigonometric functions, exponential functions, limits and continuity, differentiation, differentiation rules, implicit differentiation, inverse trigonometric functions, derivatives of inverse functions and logarithms, applications of derivatives, extreme values of functions. Recognize the chain rule for a composition of three or more functions. You can nd more examples of using the chain rule in your text book in section 3. In calculus, the chain rule is a formula for computing the. Because i wanted to make this a fairly complete set of notes for anyone wanting to learn calculus i have. Implicit differentiation in this section we will be looking at implicit differentiation. Lets solve some common problems stepbystep so you can learn to solve them routinely for yourself. Because i wanted to make this a fairly complete set of notes for anyone wanting to learn calculus i have included some material that i do not usually have time to cover in class and because this changes from semester to semester it is not noted here. For example, the quotient rule is a consequence of the chain rule and the product rule. Sep 21, 2012 finally, here is a way to develop the chain rule which is probably different and a little more intuitive from what you will find in your textbook. Brush up on your knowledge of composite functions, and learn how to apply the chain rule correctly. Also learn what situations the chain rule can be used in to make your calculus work easier.
Calculuschain rule wikibooks, open books for an open world. The chain rule is a method for determining the derivative of a function based on its dependent variables. Most students will expand the binomial to get and differentiate the result to get. Find materials for this course in the pages linked along the left. Here is a set of practice problems to accompany the chain rule section of the derivatives chapter of the notes for paul dawkins calculus i course at lamar university. The chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary. The videos, which include reallife examples to illustrate the concepts, are ideal for high school students, college students, and anyone interested in learning the basics of calculus. Roughly speaking the book is organized into three main parts corresponding to the type of function being studied. The next theorem, which we have proven using the chain rule, allows us to find. The inner function is the one inside the parentheses. In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. The chain rule if youre reading this, chances are you already know what the chain rule is and are ready to dive in. Chapter 1 is on sets, real numbers and inequalities. This gives us y fu next we need to use a formula that is known as the chain rule.
In calculus, the chain rule is a formula to compute the derivative of a composite function. In this section, we will learn about the concept, the definition and the application of the chain rule, as well as a secret trick the bracket. While our structure is parallel to the calculus of functions of a single variable, there are important di erences. From the table of contents it seems that the index pages are supposed to be in the original book. Furthermore, the index of applications at the back of the book provides students and instruc. State the chain rules for one or two independent variables. Show solution for exponential functions remember that the outside function is the exponential function itself and the inside function is the exponent. After a suggestion by paul zorn on the ap calculus edg october 14, 2002 let f be a function differentiable at, and let g be a function that is differentiable at and such that. Understanding basic calculus graduate school of mathematics. Except for the simplest functions, a procedure known as the chain rule is very helpful and often necessary to find derivatives.
For problems 1 27 differentiate the given function. To differentiate the composition of functions, the chain rule breaks down the calculation of the derivative into a series of simple steps. Chain rule the chain rule is one of the more important differentiation rules and will allow us to differentiate a wider variety of functions. All books are in clear copy here, and all files are secure so dont worry about it. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function f.
Namely, i wanted a book written by someone who actually knows how to write howto books instead of by a mathematician writing something that will make sense to other mathematicians. It was submitted to the free digital textbook initiative in california and will remain unchanged for at least two years. Calculus this is the free digital calculus text by david r. The book is in use at whitman college and is occasionally updated to correct errors and add new material. The chain rule has many applications in chemistry because many equations in chemistry describe how one physical quantity depends on another, which in turn depends on another. Calculus s 92b0 t1 f34 qkzuut4a 8 rs cohf gtzw baorfe a cltlhc q. The chain rule mctychain20091 a special rule, thechainrule, exists for di. Use tree diagrams as an aid to understanding the chain rule. We will use it as a framework for our study of the calculus of several variables. Super secret number puzzle chain rule edition find the answer to each question. The book is well written and covers both big pictures and technical details of materials in calculus.
Write your answers in the answer blanks to the left. Whenever we are finding the derivative of a function, be it a composite function or not, we are in fact using the chain rule. The derivative of sin x times x2 is not cos x times 2x. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. On completion of this worksheet you should be able to use the chain rule to differentiate functions of a function. But there is another way of combining the sine function f and the squaring function g into a single function. I was comparing my attempt to prove the chain rule by my own and the proof given in spivaks book but they seems to be rather different. Thomas calculus, twelfth edition, helps readers successfully generalize and apply the key ideas of calculus through clear and precise explanations, clean design, thoughtfully chosen examples, and superior exercise sets. This lesson contains plenty of practice problems including examples of chain rule problems with trig functions. In this example, we use the product rule before using the chain rule. Chain rule the chain rule is one of the more important differentiation rules and will. Proof of the chain rule given two functions f and g where g is di. The outer function is v, which is also the same as the rational exponent.
Recall that with chain rule problems you need to identify the inside and outside functions and then apply the chain rule. This section presents examples of the chain rule in kinematics and simple harmonic motion. To this end, i have tried to write in a style that communicates intent early in. Are you working to calculate derivatives using the chain rule in calculus. Calculus 1 class notes, thomas calculus, early transcendentals, 12th edition copies of the classnotes are on the internet in pdf format as given below. Atkhan academy note your text, khan academy and even looking up chain rule on youtube may give you examples of derivative rules we have not covered yet. Apply the chain rule and the productquotient rules correctly in combination when both are necessary.
Write the composite function in the form fgx by explicitly labeling the inner function u gx and the outer function y fu. The chain rule can be one of the most powerful rules in calculus for finding derivatives. Read online calculus cheat sheet university of georgia book pdf free download link book now. The topics include curves, differentiability and partial derivatives, multiple integrals, vector fields, line and surface integrals, and the theorems of green, stokes, and gauss. That is, if f and g are differentiable functions, then the chain rule expresses the derivative of their composite f.
For an example, let the composite function be y vx 4 37. Mit professor gilbert strang has created a series of videos to show ways in which calculus is important in our lives. However, in the current pdf version the index seems to be missing. To see this, write the function fxgx as the product fx 1gx. Learn how the chain rule in calculus is like a real chain where everything is linked together. After the chain rule is applied to find the derivative of a function fx. Example 1 use the chain rule to differentiate rzv5z. To this end, i have tried to write in a style that communicates intent early in the discussion of each. Most students will expand the binomial to get and differentiate the result. Because one physical quantity often depends on another, which, in turn depends on others, the chain rule has broad applications in physics.
The chain rule allows us to combine several rates of change to find another rate of change. It tells you how to nd the derivative of the composition a. The chain rule, part 1 math 1 multivariate calculus. Published in 1991 by wellesleycambridge press, the book is a useful resource for educators and selflearners alike. Click here for an overview of all the eks in this course.
Calculus is about the very large, the very small, and how things changethe surprise is that something seemingly so abstract ends up explaining the real world. This function has an inside function and an outside function. The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule. It is useful when finding the derivative of a function that is raised to the nth power. Differentiating using the chain rule usually involves a little intuition. In singlevariable calculus, we found that one of the most useful differentiation rules is the chain rule, which allows us to find the derivative of the composition of two functions. The right way to begin a calculus book is with calculus. The chain rule also has theoretic use, giving us insight into the behavior of certain constructions as well see in the next section. Unfortunately the rule looks a bit odd, and its unclear why it works they way it does. A few figures in the pdf and print versions of the book are marked with ap at.
The chain rule can be used to derive some wellknown differentiation rules. The substitution method for integration corresponds to the chain rule for differentiation. This book covers the standard material for a onesemester course in multivariable calculus. Work through some of the examples in your textbook, and compare your. The more times you apply the chain rule to different problems, the easier it becomes to recognize how to apply the rule. It is well organized, covers single variable and multivariable calculus in depth, and is rich with applications. For example, if a composite function f x is defined as. Students should notice that the chain rule is used in the process of logarithmic di erentiation as well as that of implicit di erentiation. If z is a function of y and y is a function of x, then the derivative of z with respect to x can be written \fracdzdx \fracdzdy\fracdydx. May 11, 2017 this calculus video tutorial explains how to find derivatives using the chain rule. The best way to memorize this along with the other rules is just by practicing until you can do it without thinking about it. Some problems will be product or quotient rule problems that involve the chain rule. Just because we now have the chain rule does not mean that the product and quotient rule will no longer be needed. Thomas offers the right mix of basic, conceptual, and challenging exercises, along with meaningful applications.
199 74 740 1313 1320 1293 1505 1546 242 1243 1313 1302 389 57 1136 1104 1534 1327 74 1076 1098 1486 814 716 428 1433 1394