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 For better work & study skills, read chapters 2  in  Three Skills for Algebra. Sooner is better. Good luck.

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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 ®
Dy
Dx
  =   lim
Dx ®
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 ®

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 
Dy
Dx
 =  

f(x+h) -f (x)
h

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 ®
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 ®

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)2-2 =  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)2-3 =  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)2-5 =  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-a =  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-x =  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 ®

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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For  help in calculus, explore
Volumes
2. Three Skills for Algebra and 3. Why Slopes & More Math, and  Calculus Introduction site area. See how to learn or teach key skills and concepts, some not all.

Calculus Videos
0.   First Calculus Preview
0. Triangle Inequality
0 Inequalities
0. Solving Inequalities
1. Distance+Midpt Formulas
1. Function Domains
1.Polynomial Domain+Range
1. Fn:  Linear Combinations
1. Fn Composition II
1. Fn Composition I
1. Solving y**n = x**m
2 .Real Numbers
2. Limits Numerical View
2. Limits,. Formal Definition
2. Limit Properties Numerically
2. Decimal View of Limits
2. Error Control View
2. Limits & Continuity
2. Limit Vals via Substitution
2. Limits & Composite Fns
2.. Limit Examples
2. One Sided Limits
2. Infinity and Limits
2. Parameters in Limits
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.  Linear Chain Rule
3. Chain Rule for Powers
3. Chain Rule - Polynomials
3. Chain Rule Examples I
3. Chain Rule Examples II
3. Linear Approximation I
3. General Chain Rule
3. Inverse Fns Derivatives
3. Chain Rule: ln(x) & exp(x)
3. Square & Cube Roots
4.  Linear Approximation
4. Second Derivative Test
4. Sketch y = x^3 - 6x^2- 12x
4. Sketch y = x^3 - 3 x^2 - 9x
4. Sketch y = 1 - 1/(1+x^2)
5. Indefinite Integrals A
5. Indefinite Integrals B.
5  Indefinite Integrals C
6. Definite Integral D
6. Area Under Curves
7. Volume of a Sphere

To Learn More, visit Volumes 2 and 3.


Advanced Topics

Limit Properties Algebraically
Pigeon Hole Principle
Bolzano
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
From Lipschitz Continuity

To Learn More: Visit Real Analysis.

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