More 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. |
Starter Guide (Views)
Real Player Videos
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
Starter & Warm Up Lessons
1. Usual Review/Starter Lessons
2. Limits [13]
3. Differentiation Rules[28]
4. Applications of Derivatives [5]
5. Definite Integrals - Preview [5]
6. Integration Applications [6]
Advanced Material
| | Sum Rule
The statement of these rules follows the video examples.
Differentiation rules say how to compute formulas for f '(x1)
in a routine mechanical manner from formulas for f(x), at least when the
formula for f(x) is simple enough. The proof, justification and further
explanation of rules for differentiation may be found in this site area.
Again, 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.
Sum Rule: If h(x) = au(x)+ bv(x) for some real numbers a and b,
then g'(x) =a u '(x) + bv'(x)
Assume a = 1 and b = 1 on first reading. The special cases
- a = b =1 gives the sum rule
d ( u + v) = du
+ dv
dx
dx dx
- a = 1 & b = -1 gives the difference rule.
d ( u - v) = du
_ dv
dx
dx dx
Proof:
|
g(x+h)-g(x)
h |
= a |
u(x+h)-u(x)
h |
+ b |
u(x+h)-u(x)
h |
|
Now taking the limit as h approaches zero yields the stated sum rule.
Elementary Example: If f(x) = 4 x + 5 and g(x) = x2
then
f'(x) = 4 and g'(x) = 2x. So h(x) = g(x) + f(x) implies
h'(x) = 2x+4
Generalized
Sum Rule:
| If |
g(x) = |
n
å
j = 1
|
aj* uj(x) |
| then |
g'(x) = |
n
å
j = 1
|
aj* u'j(x) |
The proof follows by mathematical induction on n > 1.
| |
|