125 2.2 and 2.3

2.2 and some of 2.3 Some Differentiation Formulas

A. Derivative of a ConstantB. Power RuleC. Evaluation of a DerivativeD. Leibniz NotationE. Derivatives in Business and Economics

1

A. Derivative of a Constant

• The derivative of a constant is _______.• Examples:

2

B. Power Rule

• The power comes down front with the coefficient, and you deduct one from the power. Here’s the rule:

1or

,derivative then the, If

n

n

xnxfdx

dxf

xxf

3

Power Rule

• The power comes down front with the coefficient, and you deduct one from the power. Here are examples:

4

What’s the derivative of x?

• What about x-5?

5

What’s the derivative of x1/4?

• What’s the derivative of 8x-1/2?

6

What’s the derivative of 7x?

• What’s the derivative of x3 + x5?

7

What’s the derivative of x3 - x5?

• What’s the derivative of 5x-2 - 6x1/3 + 4?

8

What’s the derivative of 4x-3 - 3x1/4 + 171?

• What’s the derivative of -4x1/2 - 6x-3 + x?

9

C. Evaluation of a Derivative

• Evaluate the derivative of f(x)= x2 for x = 3.

• Evaluate the derivative of f(x) = x for x = 3.

10

.1for 465 Evaluate 3/12 xxxdx

d

11

.2 find then , If 4 fxxf

12

3. at x curve theo tangent tis that line

theof slope thefind then 2x,3 If 4

xxf

13

Remember when your algebra teacher taught you this:

• Rewrite this so that it has no denominators:

22 3 1x x

f xx

14

The reason we need to that is …

• We don’t yet know how to differentiate

• But we know to differentiate this:

• So rewriting it will be your very first step!

22 3 1x x

f xx

15

You try differentiating:

3 2

2

6 3 2 1x x xf x

x

16

DON’T FORGET WHAT A DERIVATIVE IS

• It’s a function for the slope of the tangent line.• If you plug in a value for x. Let’s say, you find

the derivative of f and call it f prime. Then suppose you plug in x = 3 into f prime and get 4. So that f’(3) = 4. What does this mean?

• THE LINE THAT IS TANGENT TO f AT THE POINT x=3 HAS A SLOPE OF 4.

17

D. Leibniz Notation

1

3

2

find ,4 If

wroteLeibniz ,2 writingof Instead

x

x

dx

dfxxf

dx

dff

18

E. Derivatives in Business & Economics

• COST FUNCTIONS:• C(x) is a function for “total cost of producing x units.”

• MC(x) is for “MARGINAL COST” and it is the same as C’(x), the derivative of C(x).

• Marginal Cost gives you COST PER UNIT.

19

REVENUE FUNCTIONS:

• R(x) is a function for the total revenue from selling x units.

• MR(x) is for “MARGINAL REVENUE,” and it is the same as R’(x), the derivative of R(x).

• Marginal Revenue gives you REVENUE PER UNIT.

20

PROFIT FUNCTIONS:

• P(x) is a function for the total profit from producing and selling x units.

• Profit = Revenue minus Cost• P(x) = R(x) – C(x)

• MP(x) = P’(x) = Marginal Profit = Profit per item

21

answer.your interpret and made,

been have s telephone100cost when marginal theFind

MC(x). Find dollars. 500400)(

is nes) x telephomaking ofcost total(thefunction cost its

finds and s telephonecordless esmanufacturcompany A

xxC

22

f(8)? from meaningin differ that does How

answer.your interpret and minutes 8after function this

of IRC theFind .6 is minutesin x memorize

canperson a that names of # that thefinds researcherA

3 2xxf

23

mean? R(20) doesBut what

answer.your interpret and sold,been

have chairs rocking 20 when revenue marginal theFind

MR(x). Find dollars. 23)( isfunction revenue

its finds and chairs rocking sellscompany A :You try2xxR

24