2013-12-02 product rule, why you should believe it

7/27/2019 2013-12-02 Product Rule, Why You Should Believe It

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NAME DATE BAND

PRODUCTRULE: WHYYOUSHOULDBELIEVEIT

CALCULUS| PACKERCOLLEGIATEINSTITUTE

We learned and by learned, I mean your teacher told you that if you have the productof two functions, ( ) ( )y f x g x= , then we know the derivative is:

' '( ) ( ) '( ) ( )y f x g x g x f x= + Now lets think about what this means. We are saying that

the rate of change of the entire product

!the rate of change of the "st f#n$!the %nd f#n$&!the rate of change of the %nd f#n$!the"st f#n$

I hope to give you three waysto understand it. First, a conceptualreason why youshould believe its true. Second, some concrete e#amples to show you how it alreadyworks with what you know. 'nd third, an algebraic proof using the formal definition ofthe derivative.

Section 1: A conceptual understanding

(et us imagine a scenario where a product is a natural thing. (ets imagine that you areinvesting in a stock which is changing its price over time. (et ( )f t be the number of

shares of stock you have at time t, and let ( )g t be the price of each share at time t.

)he total value of your portfolio at time tis *****************************.

(ets imagine you want to see how the value of your stock is changing over time. )here

are two thingsthat affect it. +ou have the number of shares of stock you have, which canincrease or decrease over time depending on if you buy or sell. 'nd you have the price ofthe stock, which changes with the market.

(ets imagine you are investing in 'pple.

+ou buy -- shares of the stock at //0.

)he total value of your portfolio is

****************************.

' minute later, you reali1e you want 2ust a bit morestock. +ou buy "- more shares of stock. 3oweverthe price of the stock is now /0-. )he total value ofyour portfolio is

****************************.

Now remember what our goal was4 it was to find the rate of change of the value ofyour portfolio. I am going to argue, graphically, that the rate of changeof the value ofyour portfolio is going to be:

!the 567 of the number of shares of the stock$!the price of the stock$&

!the 567 of the price of the stock$8!the number of shares of stock$

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&a tiny tiny tiny tiny tiny bit that wont really matter

(ets look at whats going on graphically4

't the beginning4 6ne minute later4

Now look at '( ) '( ) ( ) '( ) ( )ValueOfPortfolio t f t g t g t f t = +

)hat tiny tiny tiny amount is something we wont worry about. 9or derivatives, we aretalking about taking two points which are infinitessimalyclose to each other, meaningthe time interval is infinitely short. 5ight now our interval is " minute. It turns out thatwhen you look at things rigorously !which we will do with the proof below$, that the tinytiny tiny amount will go to 1ero. (imits, and all that.

)his is nota rigorous proof. ut it suggests in a conceptual way why the derivative ofthe product of two functions has the form it has.

Section 2: Concrete Examples

We are going to find a few derivatives both withand withoutthe product rule. +ou willsee that the product rule matches what we e#pect it to be.

9ind the derivative withoutthe productrule

9ind the derivative withthe product rule!underline the first part of the product andthe second part of the product$

( ) ( )5 2( ) 5a x x x= ( ) ( )5 2( ) 5a x x x=

( )b x x x= ( )b x x x=

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( ) 2 xc x e= ( ) 2 xc x e=

3 5( ) (2 )( )d x x x x x= + 3 5( ) (2 )( )d x x x x x= +

2( ) 6j x x= 2( ) 6j x x=

( ) (2)(6)k x = ( ) (2)(6)k x =

3opefully you see for the functions above that the product rule works. 'nd hopefullyyou see that for these functions, the product rule isnt faster or more useful than whatwe already know.

;o why might we want the product rule< )ry to find the derivative of the functions belowwithoutthe product rule and youre in trouble= >o you see why< So use the productrule and find the derivatives

3( ) ( 2 ) xm x x x e= + ( ) x xn x e e=

2( ) sin( )p x x x= ?and Im letting you know

that the derivative of sin( )x is cs( )x @( )

2

( ) sin( )q x x=

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Section !: An Alge"raic #roof

9or the following, write down the reason you are allowed to do each step. e as clear aspossible.

(et ( ) ( )y f x g x= .

)hen to find 'y we need two points infinitely close to each other. )hey are:

************************ and **************************.

)hus 'y =

!

!

!

!

!

( ) ( " ) ( ) ( )#i$

( ) ( ) ( ) ( ) ( )#i$

( )% ( ) ( )& ( )i$

( )% ( ) ( )& ( )&

( ) ( ) ( )

( )% ( )

( )#i$

( )% ( ) ( )&

% ( )

#i$

h

h

h

h

h

f x h g x f x g x

h

f x h g x h f x h f x g x

h

f x h g x h g x f x

h

f x h g x h g x f

h

g x f x h g x

g x f x h

g x f x h x

h h

f x h g x h g x

h

+ +

+ +

+

+

+ + +

+ +

+ +

+

+

+

+!

! ! ! !

! !

( )% ( )

% ( ) ( )

% ( ) ( )

( ) % ( ) % ( )&

*i+,#

( )i$

% ( ) ( )& i$ ( ) #i$ #i$ ( ) #i$

% ( ) ( )& &( ) #i$ ( )

n-#./ % ( )& ( ) %

#i$

(

& ( )

h

h h h h

h h

f x

h

g x h g xf x h g x

h h

g x

g x f x h

f x h f x

h g xf x g x

h h

f x h f x

d df x g x f x

dx dx

f

g x

d d

dx dxx g x g x

+

+ +

+

+

+

++

+

+

) & ( )f x

0

S*$ *# #i$i-s (- #i$i- , s*$ is - s*$ - #i$i-s)

P*c- *# #i$i-s (- #i$i- , 4*c- is - 4*c- -

#i$i-s)


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#ractice with the #roduct $ule %raphically&Conceptually

". elow is a graph of ( )p x and ( )q x . (et ( ) ( ) ( )r x p x q x= , which is notgraphed.

!a$ '(!) p = and '(!) q =

!b$ '(!)r =

!c$ '(2)r =

!d$ '( 25)r =

!e$9ind the eAuation of the tangentline to ( )r x at 2x = .

%. ;uppose '(2) 07 '(2) 37 (2) 17f g f= = = and (2) 1g = . 9ind the derivative at 2x = of thefollowing functions:

!a$ ( ) ( ) ( )s x f x g x= +

!b$ ( ) ( ) ( )p x f x g x=

!c$ 9ind the eAuation of the tangent line to ( )p x at 2x = .

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B. (et ( ) ( ) ( )h x f x g x= where the graphs of ( )f x and ( )g x are below.

!a$Cvaluate ( 2)h and (3)h .

!b$Cstimate '( 2)7 '(3)7 '( 2)7 '(3)f f g g

!c$Cvaluate '( 2)h and '(3)h .

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2013-12-02 product rule, why you should believe it

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