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Limit Definition of Derivatives
and x-dependence
[Play Video]
3½ minutes: Three Notations for derivatives, prime, functional or
Liebniz y' = y'(x) = dy/dx
The previous limit motivation page suggested
| f '(x1) |
= |
lim
Dx -> 0 |
|
Dy
Dx |
|
|
= |
lim
Dx -> 0 |
|
f(x1+Dx)
-f (x1)
Dx |
|
as the definition of the slope and derivative of f(x) at the point x = x1.
Change of Notation: In the following discussion, to make my typing
easier, we will drop the subscript on x, so x1 become x,
and replace Dx by h.
Definition
The slope m to a curve y = f(x) at x is defined by the limit calculation
| f¢(x)
= |
lim
h -> 0 |
|
f(x+h) -f (x)
h
|
|
Notes: The right hand side of the above equation may be read as the limit as h approaches zero of the
quotient and slope of secant lines
For each point x = a, the value of f
'(a) when defined provides the slope of the tangent line through the point [a,f(a)]
or in brief at the points x = a on the curve.
Some may write
| f¢(x)
= |
lim
Dx -> 0 |
|
Dy
Dx |
|
instead, as a change of notation. However, typing h is easier that typing
Dx . So these online lessons
favour h.
Reference: Limits with Parameter
In the calculation of the right hand side,
| f ¢(x)
= |
lim
h -> 0 |
|
f(x+h) -f (x)
h
|
|
the limit process eliminates the dependence on h and leaves the dependence on
x. The latter appears as a parameter in this calculation. Elimination
of h via the limit process results in a function f '(x). We will see below how
the formula for f '(x) can be obtained (derived) from the formula for f(x). For
that reason, f '(x) could be called (but is not) the obtainable instead of
the derivative of f(x)
Algebraic Formulas for f '(x) from formulas for f(x).
Here the calculation of f '(x) give a function. If you understood, the
calculations of Limits with Parameters
and how the results depend on the parameter or variable x. then the following examples
may be too repetitive. But the repetition is intended to suggest a pattern and
to develop or confirm your algebraic reasoning skill.
Consider the function y = f(x) = x2
(again) We will compute the slope, that is the derivative of this function at x
= 2, x = 3, x = 5 and x = a. Look for a pattern in
the following arithmetic computations.Example 1. Let x = 2.
Then
f(2+h) - f(2) = (2+h)2-22 = 22
+ 2(2)h + h2 = 2(2)h + h2 =h(2(2) + h)
Therefore
| f '(2) = |
lim
h-> 0 |
f(2+h)-f(2)
h |
= |
lim
h-> 0 |
h(2 + h)
h |
= |
lim
h-> 0 |
2(2) + h
|
= 2(2)
|
Of course 2(2) = 4, but for the sake of pattern recognition and emphasis, we
keep the arithmetic expression 2(2) to the end of the calculation.
Example 2. Let x = 3. Then
f(3+h) - f(3) = (3+h)2-32 = 32
+ 2(3)h + h2 = 2(3)h + h2 =h(2(3) + h)
Therefore
| f '(3) = |
lim
h-> 0 |
f(3+h)-f(3)
h |
= |
lim
h-> 0 |
h(3 + h)
h |
= |
lim
h-> 0 |
2(3) + h
|
= 2(3)
|
Of course 2(3) = 6, but for the sake of pattern recognition and emphasis, we
keep the arithmetic expression 2(3) to the end of the calculation.
Example 3. Let x = 5. Then
f(5+h) - f(5) = (5+h)2-52 = 52
+ 2(5)h + h2 = 2(5)h + h2 =h(2(5) + h)
Therefore
| f '(5) = |
lim
h-> 0 |
f(5+h)-f(5)
h |
= |
lim
h-> 0 |
h(5 + h)
h |
= |
lim
h-> 0 |
2(5) + h
|
= 2(5)
|
Of course 2(5) = 6, but for the sake of pattern recognition and emphasis, we
keep the arithmetic expression 2(5) to the end of the calculation.
The Common Algebraic Pattern
The three examples follow the same pattern. We will rewrite the above
calculations with the letter a replacing the numbers 2, 3 and/or 5 above,
to emphasize the pattern. In the rewrite below, note that the role of a
below could be played or assumed by each of the numbers 2, 3 or 5 above, another
number or even the letter x..Example n. Let x = a.
Then
f(a+h) - f(a) = (a+h)2-a2 = a2
+ 2ah + h2 = 2ah + h2 =h(2a + h)
Therefore
| f '(a) = |
lim
h-> 0 |
f(a+h)-f(a)
h |
= |
lim
h-> 0 |
h(a + h)
h |
= |
lim
h-> 0 |
2a + h
|
= 2a
|
Last Example. Let x = x. Then
f(x+h) - f(x) = (x+h)2-x2 = x2
+ 2xh + h2 = 2xh + h2 =h(2x + h)
Therefore
| f '(x) = |
lim
h-> 0 |
f(x+h)-f(x)
h |
= |
lim
h-> 0 |
h(x + h)
h |
= |
lim
h-> 0 |
2x + h
|
= 2x
|
In the last example, we see that the derivative of f(x) = x2 is
2x. Here again the limit process eliminates the h-dependent of the slope
quotient and leaves the x dependence.
Video Examples
(REDUNDANT?)
[Play
Video] 2¼ minutes: Derivative of x3
algebraically via Limits.
[Play
Video] 2¼ minutes: Derivative as a Limit of a Quotient. First
pass at finding the derivative or slope of f(x) = x2.
Algebraic View.
[Play
Video] 2¼ minutes: Second pass at finding the derivative
or slope of f(x) = x2 at two values of x. Numerical
Examples of Limit Evaluation to suggest a pattern.
Going Further - A Look Ahead
The formula or definition
| f¢(x)
= |
lim
h -> 0 |
|
f(x+h) -f (x)
h
|
|
provides the initial limit-based way to compute f '(x). We will use it
to obtain derivatives of simple functions. But we will also introduce
rules of differentiation which permit the calculation of formulas for f '(x)
from formulas for f(x), calculation shortcuts for the evaluation of the limit
definition of f'(x).
Here we see the plan, namely a quantity is represented by a limit. Then rules
are developed to evaluate the limit directly or replace the limit evaluation by
an equivalent calculation in which there is no mention of limits. But the basic
properties of all these calculations come from limit considerations.
Video Example
[Play
Video] 2¼ minutes: Derivative of x3
algebraically via Limits.
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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.
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volumes 2 and 3 before calculus and during it. |
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For Parents & Teachers: Speaking
Skills, Reading
& Writing,
Preparing for Science, ends,
values and methods for work and study, parent- friendly mathematics booklets for ages 4-14.
-
Math
Education
Essays (opinions,
possibilities, references)
- POMME, a two
level program for future skill development in
schools and colleges worldwide. Address content &
motivation gaps with ends, values & methods for skill
development to say which way to go, how and why. -
Present Day Curriculum:
(A) Secondary
I Mathematics
consolidate fractions and measurement, skills and
sense consolidation,
(B)
Secondary II Mathematics
year of algebra and proportionality
(C) See too the following:
- Arithmetic
& Number Theory Practices (horribly put, but
useful)
- Algebra and
Logic SubProgram
(well put, extremely useful)
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.
Calculus Prep or Help: See Volumes 2 & 3,
and this bigger
Calculus
Guide.
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Senior
High School &
Calculus Students
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Free Live Lesson
- Operations with Decimals - Comparison, Subtraction and Long Division
- Click here
to attend.
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For Senior
High School Mathematics & Calculus
Intro to Solving
Linear Equations
- a different paths for junior and even senior high
school students.
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)
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Many 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
Use Forward & 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)
- POMME, a two
level program for instruction K1-14
- 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).
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Skill Development Tips
For All
Standards: (A) Take
care to avoid the domino effect of errors & approximations; (B) Do and
record steps in an manner that allows skill mastery to be seen or
corrected. Anything represent substandard work.
Key Numerical Methods
- To multiply signed numbers, prefix the product of their signs to the product
of their lengths or unsigned parts. The product is negative if the no of
negative sign in it is odd.
- To add signed numbers with like signs, prefix the common sign to the sum of the
lengths.
- To add signed numbers with opposite signs, prefix the sign of the longest to
the difference: length of longest minus length of shortest.
- Should we study roots and powers of real numbers with formulas involving exponential and log.
- How does adding and multiplying points in the plane and rotating the midpoint
of a line segment lead to mastery of complex numbers and the thought-based
development of their properties, all before trig?
- New Axioms for High School Mathematics: In accounting, totals of assets
and debts may be calculated by dividing the assets and debts into
non-overlapping (disjoint) groups and then adding subtotals. In general, sums
(and products) of counts and numbers, positive and negative numbers
included, can be obtained by adding subtotals (and multiplying
subproducts, respectively). These practices may be cast as axioms in
secondary mathematics. Then operations on polynomials are easily implied
justified by these "axioms" and the geometric introduction of column
methods for expanding a products of two sums. While set theory in pure
mathematics may imply the above axioms in university mathematics programs
instruction, an earlier and more accessible explanation based on easily accepted
and understood geometric and counting practices derivation of the above
axioms is possible at the high school for students heading for college programs
in science.
In Volume 2: Prep for Calculus
- What is the difference between saying A if B and saying A if and
only if B. Being aware of the difference will sharpen ye wits.
- What is a chain of reason?
-Are your arithmetic skills OK?
-Have words been missing in the introduction of algebra?
- Can ye talk about numbers & quantities varying apart from or before the
use of letters & functions?
- Do ye know about the forward & backward use of formulas?
-Contrapositive: is that backward use of A if B?
-What is a variable x? Answer before speaking of function f(x) = x.
-What a twist! There are no rules of algebra for subtraction and division. But
if you replace them by addition of -x and multiplication by 1/x, rules of
algebra (properties of arithmetic) can be used.
In Volume 3: Calculus Slowly?
-Why are slopes studied and polynomials factored in high school?
- Volume 3 suggests how to ease or delay algebra shock in
calculus *& beyond. In Calculus, derivatives and integrals introduced and defined by limits, but calculated
without when possible by using differentiation rules forwards and backwards. The
second site calculus section may help in differential calculus.
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