Chapter 7

RS - Ch 7 - Rules of Differentiation

Chapter 7

Rules of Differentiation

& Taylor Series

Isaac Newton and Gottfried Leibniz

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(for private use, not to be posted/shared online)

7.1 Review: Derivative and Derivative Rules

? Review: Definition of derivative.

f ?x0 ? ?x ? ? f ?x0 ? dy

?y

? lim

?

?x?0 ?x

?x?0

dx

?x

f ' ( x) ? lim

? Applying this definition, we review the 9 rules of differentiation:

0

1) Constant:

??

2) Power:

3) Sum/Difference

4) Product

?

? ?

? ?

2

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RS - Ch 7 - Rules of Differentiation

7.1 Review: Derivative and Derivative Rules

? (continuation) 9 rules of differentiation:

/

5) Quotient (from 4)

? ?

? ? /? ?

??

6) Exponential

?

7) Chain Rule

(with ?

? ?

? ? be a strictly monotonic function.

.

8) Inverse function. Let ?

9) Constant, Product and Power (from 1, 2 & 4)

???

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7.1.1 Constant Rule

? Recall the definition of derivative.

f ?x0 ? ?x ? ? f ?x0 ? dy

?y

? lim

?

?x?0 ?x

?x?0

dx

?x

f ' ( x) ? lim

? Applying this definition, we derive the constant rule:

The derivative of a constant function is zero for all values of x.

dy

d

k ?0

?

dx

dx

dy

f ( x ? ?x) ? f ( x)

? f '(x) ? lim

?

x

?

0

dx

?x

If f(x) ? k

then f(x ? ? x) ? k

k?k

lim

? lim 0 ? 0

?x ? 0

?x ? 0

?x

y ? f ?x ? ? k

?

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RS - Ch 7 - Rules of Differentiation

7.1.2 Power-Function Rule

( x ? ��x ) n ? x n

f ? x ? ��x ? ? f ? x ?

? lim

?x ? 0

��x ? 0

��x

��x

n

n ?1

n?2

2

( x ? nx ��x ? ( n ? 1) x ��x ? ... ? nx��x n ?1 ? ��x n ) ? x n

? lim

?x ? 0

��x

n

n ?1

n?2

? lim x / ��x ? nx ? ( n ? 1) x ��x ? ... ? nx��x n ? 2 ? ��x n ?1 ? x n / ��x

f '(x) ? lim

?x ? 0

? nx n ?1

lim

��x ? 0

( x ? ��x ) n ? x n

��x

? nx n ?1

Example: Let Total Revenue (R) be:

R = 15 Q �C Q2

?

??

15

2?.

As Q increases R increases (as long as Q > 7.5).

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7.1.3 Sum or Difference Rule

d

? f ?x ? ? g ?x ?? ? f ??x ? ? g ??x ?

dx

? The derivative of a sum (or difference ) of two functions is the

3)

sum (or difference ) of the derivative s of the two functions

Example :

C ? Q 3 ? 4Q 2 ? 10Q ? 75

d

d

d 3 d

dC

?

4Q 2 ?

10Q ?

75

Q ?

dQ

dQ

dQ

dQ

dQ

dC

? 3Q 2 ? 8Q ? 10 ? 0

dQ

6

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RS - Ch 7 - Rules of Differentiation

7.1.4 Product Rule

?

4

? ?

? ?

The derivative of the product of two functions is equal to the

second function times the derivative of the first plus the first

function times the derivative of the second.

Example: Marginal Revenue (MR)

Total Revenue: R = P Q

Given ?

15

?

?

? R

15

?

?

?

? ?

1 ? 15

?

15

2?

Same as in previous example.

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7.1.5 Quotient Rule

5

? ? ? /? ?

??

??

? ?

??

??

? ? /? ?

??

Example :

TC ? C(Q)

AC ? C(Q)/Q

Total cost

Average cost

d C ?Q ? Q ? C ??Q ? ? C ?Q ? ?1 1 ?

C ?Q ? ? 1

?

? ?C ??Q ? ?

? ?MC ? AC ?

2

dQ Q

Q

Q?

Q ?? Q

d C ?Q ?

if

? 0, then AC ? MC (Average Cost ? Marginal Cost)

dQ Q

8

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RS - Ch 7 - Rules of Differentiation

7.1.6 Exponential-Function Rule

f '(x) ? lim

?x ?0

f ?x ? ��x ? ? f ?x ?

e? ( x ? ��x ) ? e?x

(e?��x ? 1)

? lim

? e?x lim

��x ?0

?x ?0

��x

��x

��x

(e h ? 1)

?1

h ?0

h

Definition of e : e unique positive number for which lim

(e h ? 1)

?��

h ?0

h

Let h ? �ʦ�x. Then, lim

Thus, lim

��x ?0

e? ( x ? ��x ) ? e?x

? ?e?x

��x

? Example : Exponential Growth

d 0.5t

y ? f ?t ? ? e 0.5t

e ? 0.5e 0.5t

dt

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7.1.6 Exponential-Function Rule: Joke

?

?

?

?

A mathematician went insane and believed that he was the

differentiation operator. His friends had him placed in a mental

hospital until he got better. All day he would go around frightening

the other patients by staring at them and saying "I differentiate you!"

One day he met a new patient; and true to form he stared at him and

said "I differentiate you!", but for once, his victim's expression didn't

change.

Surprised, the mathematician collected all his energy, stared fiercely

at the new patient and said loudly "I differentiate you!", but still the

other man had no reaction. Finally, in frustration, the mathematician

screamed out "I DIFFERENTIATE YOU!"

The new patient calmly looked up and said, "You can differentiate me all

you like: I'm ex."

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