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www.whyslopes.com >  Calculus Guide >   3. Chain Rule - Step I     [ Back ] [ Up ] [ Next ]

Chain Rule for Linear Functions

In essence,  the chain rule for nonlinear is a limit and linear-approximation based generalization of the chain rule for linear functions. You could go directly to the statement of the chain rule and its proof.  However, to make the statement, application and proof of the chain rule more accessible, we will the follow the longer route of algebraically derive the chain rule for composite functions in the case where the outer function is linear, than a power and then a polynomial. This longer route will develop or reinforce your command of mathematical induction, and  make the statement of the chain rule and the composition of functions easier to grasp or clear by taking small steps.  You may compare this longer to learning to swim by wading in the shallow end and learning to glide as a preliminary step to learning to swim.   The quicker route,  hard for many, is to jump in the deep-end to sink or swim.  

Tangent Line Equation

When a skier is located on a curve y = f(x) at (x1,y1) = (x1,f(x1)), the slope of his or her ski is assumed to lie on a tangent line. This tangent line has (or is now assumed to have) the equation
y = mtangent(x-x1)+y1
where mtangent = f¢(x1) is given by a limit L discussed above. The foregoing represents the mathematical definition of the tangent line to curve y = f(x) at (x1,y1) = (x1,f(x1)).


The linear function
y = mtangent(x-x1)+y1
where mtangent = f¢(x1) = mski provides an approximation to the value of y = f(x).  

This recall of the tangent line and its equation as a linear approximation to y = f(x) is to suggest or to  the plant the idea that chain rule in general is only a limit- and approximation-based refinement of the chain rule for linear functions. 

 

Chain Rule for Linear Functions in slope notation.

The graph of  y  = a x + b = g(x)  versus x gives a line with slope a. The graph of z =  cy + d = f(y) versus y gives a line with  slope c. Then  the substitution y  = ax + b into z = cy+d yields a linear function of x whose graph is a line with slope ca. Reasons why follow.

z = cy + d  
  = c(ax+b)+d 
  = cax +cb+d 
  = m x + k

where k = cb+d and m  = ca is the slope of the graph of z versus x.

Digression:    z = f(y)  = f(g(x) = f o g (x) = m x + k  = ca x + k.    In the case where f and g are inverse functions,   mx+k = 1 and hence the product of the slopes of f and g=f -1 is  ca = m =1.
                 

Chain Rule for Composition of Linear Functions in function and derivative notation -

Chain Rule for Composition of Linear Functions

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Optional Reading: 

The chain rule for linear function could (pure speculation on my part) come from the industrial revolution and the distance traveled by chains or pulleys or gears when coupled together. 


In the above examples, three disks are connected by bands which which assume do not slip. As the first band moves without slipping, the distances traveled by A and B around their respective circles are equal, denote that common distance by S. Likewise, As the second  band moves without slipping, the distances traveled by C and D around their respective circles are equal. Denote that second distance by T.   A gear ratio for this pulley system is given by  S/T or its reciprocal.

Let  X, Y and Z denote the angle displacement of A, B and D, respectively, in radians. Then the angle displacement of C is y as well. Now, 

arclength 
(traveled)		  =  radius* radian measure
of angle

Therefore

  • S =  aX from the  circle with point A,
  • S = bY from  the circle with point B. 
  • T =  cY from the  circle with point C,
  • T =  dZ from the  circle with point D,

Therefore Y = (a/b) X and Z = (d/c)Y. So substitution implies  Z = (d/c)(a/b) X

Note the foregoing is speculation - I have not studied the linguistic or physical origins of the term chain rule in calculus. In days gone by, pioneers in a discipline would create explanations because they sound plausible and state them authoritatively, without verification.   The above speculation is given tentatively.  One day, if I remember, I will try to confirm or refute or refine this speculation.  

 

Calculus Guide

Section Entrance
Real Player Videos
My First Steps
About Calculus
1. Regular First Steps
2. Limits [13]
3. Differentiation Rules[28]
4. Applications of Derivatives [5]
5. Definite Integrals - Preview [5]
6. Integration Applications [6]
Advanced Material

Up
3. Derivative Motivation
3.  Derivative Definition I
3. Derivatives Definition II
3. Calculus: Why Radians
3 d/dx of sin(x) & cos(x)  (I)
3 d/dx of sin(x) & cos(x) (II)
3.Sum Rule
3. Product Rule
3. Power Rule
3. Previous Rules Combined
3. d/dx for Polynomials
3. Reciprocal Rule
3. Reciprocal Law: sec & csc
3. Reciprocals & Power Rule
3. Power Law for Integers < 0
3. Quotient Rule
3. Quotient Rule Examples
3. Quotient Rule: tan & cot
3.  Chain Rule - Step I
3.  Chain Rule - Step II
3.  Chain Rule - Step III
3.  Chain Rule - Step IV
3.  Chain Rule - Step V
3.  Chain Rule - Step VI
3.  Chain Rule - Step VII
3.  Chain Rule - Step VIII
3. Inverse Fns Derivatives
3. Chain Rule: ln(x) & exp(x)
3. Square & Cube Roots

Up

Reference Material: - Light reading for calculus.

Vol 2, Three Skills for Algebra covers many  topics in algebra and logic that students starting calculus should have mastered or will have to master sooner or later. Also includes arithmetic review problems to catch common mistakes.  

 Vol. 3, Why Slopes & More Maths, gives starter lessons for differential and integral calculus.
Suggestion: Read both volumes 2 and 3 before calculus and during it.


Teachers & Tutors:  See if this algebra & logic program (well put) & these Arithmetic/Number Theory Practices help. Both are prequels to POMME - a two level program for primary, secondary & even college instruction in mathematics.  Attend my live lessons just to see what is possible online.  Bon Appetit.

Senior High School  & 
Calculus Students

?
  //         \ \  
  <| (o)   (o)   |> 
 \     | |      / 
\___ _/
||
 -/[]\- 
||
   / \_ 
 What is the domino effect of errors or gaps in  figuring,  reasoning or
skill development

The Roman alphabet has 26 letters, all needed to read and write.  Arithmetic has addition, comparison, subtraction, multiplication and division of numbers & amounts.  All are needed in daily life and in higher mathematics.  

For Avid Readers in School & Out - Online Books 
   1.  Elements of Reason. 1996 
1A. Pattern Based Reason  1995 
1B. Math Curriculum Notes 1996 
2. Three Skills for Algebra  1995 
3.Why Slopes & More.Math 1995
Tour their  forewords.   

For difficulties in Algebra, Three Skills for Algebra,  in Chapters 8 to 14 & 18 etc, puts words before symbols to enrich the comprehension of all.  Those lessons form the middle part of a larger algebra (and logic)  program 

Calculus Prep or Help: See Volumes 2 & 3, and this bigger Calculus Guide.  If your  calculus   questions is not answered here, submit it. Over time, that may complete the site development of calculus. 

For Parents: Speaking Skills, Reading & Writing,  Preparing for Science,  ends, values and methods for work and study,  parent- friendly maths skill development booklets for ages 4-14.

Mostly For High School

Intro to Solving Linear Equations 
- a different paths for junior and even senior high school students. Question for Tutors: When do you use and when you skip the stick diagram method here?

Fraction Skills,  thought-based  development, Ages 10 to 14 may need a tutor.  Students who have to understand in order to do may like the development in all or part. 

For Senior High School Mathematics & Calculus

5 wordy Logic Chapters
4 curious Algebra Chapters
Words before & besides symbols. A Key Algebra forward & backwards Chapter    

First Calculus Preview (1st intro)
Four Calculus Chapters  (2nd intro)
Intro to Complex Numbers (long)
Intro to Mathematical Induction (romantic & wordy at first)

Tutors & Instructors: These lessons introduce skills differently Would you recommend them? 

More Topics 

 1. Decimal Arithmetic  Reference!
2. Integers - Intro to Signed No.s
3.  Fractions - fully explained.
4.  Fractions  with Units  
5.   Number Theory, 
6.    Solving Linear Equations  
7  Formulas for- & backwards -  
8.  Proportionality, Back- & For-wards.   
9. Logic Chapters:   
10.  Euclidean-Geometry  
11.  Slopes & Equations of Straight Lines.  (Take I. See take II below)
12.  Why Study Slopes. 
13. Maps, Plans,  Similarity & Trig,  
  (Take II included here)
14.  Quadratics: Starter lessons
15.  Polynomials: Starter lessons 
16 Why Factor Polynomials:  
17   Functions - Forwards & Backwards.  
18.  Exponents, Radicals & logs.  
19. Complex Numbers before trig (new advance/ starter lesson)
20.  DC Electric Circuits Etc 
21. Real  Analysis 
22. The Olde Complex No, Trig
& Vector Section.
23. More Calculus Stuff
- written after Volumes 2 and 3.

More For Instructors
- Education Essays   (opinions, possibilities, references) 
- Free Advice and Directions
- Math & Logic  How-TOs 
1. Arithmetic
2. Algebra
3. More Algebra
4.  Beginner Geometry
5.  More Geometry
6. Calculus 
7. Show Work or Logic 
 These may be too dense for students.

Level I Material: New Stuff
Time and Date Matters
 Level I Arithmetic. 
Money Matters
Measurement Matters
Matters of Chance (Risk Control)
 Logic Chapters (leave what's not clear in Level I to Level II)
Using/Making Maps and Plans.
(A variant of Maps, Plans,  Similarity & Trig,  to appear here).

 



 

www.whyslopes.com >  Calculus Guide  >   3. Chain Rule - Step I     [ Back ] [ Up ] [ Next ]

Road Safety Message   Walk on a side walk. If that is not possible, try  not to  walk on a road with your back to the traffic.
Try to see what  trucks, cars, buses or bicycles are coming, so that you may step out of their way.  Put safety first. .

Support for Technical Mathematics from Number Theory to Calculus Prep
A. More Arithmetic a must for algebra etc D. Logic In Mathematics G. Algebra with Take Home Value I. Vectors & Functions
Decimal Lesson - Reference  
Counting & Addition   (8 lessons)
Comparison to Subtraction  (9 lessons)
Multiplication ( 11 lessons)
Long Division  (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites 
-Primes Factorization
-Greatest Common Divisors & Multiples. 
-Prime Factorization Aids  (Learn how to find factors quickly)
-Prime Factorization Examples 
-Counting & Generating. Factors. 
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
 Fractions (< 20 lessons)  Essential Skills & Concepts 
Ratios & Fractions (3 lessons):  Similarities & Differences   
Units in calculations Fractions  with Units
B.  Basic Algebra
Solving Linear Equations  
- in one unknown. Intro  with stick diagrams?
the normal way  & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of  word problems.
  - still in more unknowns:  Gaussian Elimination via substitution, by equality or comparison, by operations on equations. 
C. More Algebra
Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra  
What is a Variable.  The answer here  is a simple prequel to the modern mathematics viewpoint.
First, every rule & pattern U meet in math, logic & science will be used forwards and backwards.  Get a head start with this theme by reading  Chapter 14 in Three Skills for Algebra.  Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial  backward  use of the proportionality formula.
 Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.  
If you can not read or write precisely, you will have difficulty in following instructions.  One wordy remedy  is given by chapters 2 to 5  in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in  Euclidean-Geometry    In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.
E.  More Geometry
The Pythagorean Theorem. Chapter 17 from  in Three Skills for Algebra uses algebra and geometry   to show why the  Pythagorean equation  for right triangles holds. Its forward and backward use  is common exercise..  At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be  fractional multiples of a unit length. That geometrically implies a  need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with  Maps and Plans drawn to scale  give coordinate-dependent base  for senior high school development of similarity, trig, vectors and straight lines.   
Complex Numbers: This lesson on Complex Numbers  draws on Euclidean and Analytic geometry. Sbortcuts simplifiy  trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products. 

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This  complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry,  personal finance, and calculus. To learn more, if you study calculus,  see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters
  1. Geometric Sums Etc,
  2. Notation For Sums,
  3. Personal Money Maths and
  4. Some Finite Mathematics
identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry:   -  Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one.  I may combine them later.  -In my school days, slopes appeared year after year.   This Why  Slopes calculus preview on graphs of functions y = f(x) explains why.  Enjoy.
Quadratics and Polynomials: -  Operations on Polynomials: Meet a light and ultraquick geometric introduction to  multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials.    Compare Fractions  with Units. Enrichment: A Plus:  The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic. 
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials  high school mathematics has technical value as part of the senior high school mathematics preparation for calculus.  This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials,  their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.  		Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms.  Vectors and their properties appear in physics, its mathematical description and formulation. 
 - Functions - Forwards & Backwards.  Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics.  comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic. 

J. Pre-Calculus Skill Check
-  Arithmetic Skill Check.  In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.  
-  All the skills and concepts in  Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with  your other duties.  Good luck.

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