,calculus ,a1 ,lab #4

  ,at ? /age 9 \r que/ 6-pute !
derivative ( e possi# func;n1 we c h&le
p[]s ( x, & we "k h[ 6di6]5tiate func;ns
built up f p[]s ( ;x us+ a4i;n1
subtrac;n1 & multiplic,n 03/ants4 ,t is1
we "k t if ;c is a 3/ant1 !n
  ?d_/dx#(cf(f)) .k c*f'(x)_4
,we al "k t
  ?d_/dx#(f(x)+g(x)) .k f'(x)+g'(x)
& t
  ?d_/dx#(f(x)-g(x)) .k f'(x)-g'(x)
if bo? ! derivative on ! "r h& sides (
^! =mulas exi/4
  ,! purpose ( ? lab is 6see if we c f9d
& def5d simil> =mulas =
  ?d_/dx#(f(x)g(x))
&=
  ?d_/dx#(?f(x)_/g(x)#)_4
  #1_4 ,a natural 3jecture bas$ on !
=mulas =! derivative (a sum or di6];e (
#2 func;ns wd 2 t
  ?d_/dx#(f(x)g(x)) .k f'(x)g'(x)_4
                                      #j
  ,n[1 we alr "k h[ 6-pute ! derivative
( k(x) .k cg(x) 0us+ ! 3/ant multiple
=mula4 ,on ! o!r h&1 k(x) .k f(x)g(x) )
f(x) .k c_4 ,is ! "kn derivative
k'(x) .k cg'(x) 3si/5t )! 3jecture t = e
;f & g, ?d_/dx#(f(x)g(x)) .k f'(x)g'(x)
_8
  #2_4 ,if f(x) .k g(x) .k x, !n we c
-pute ! derivative ( f(x)g(x)_4 ,d s4
,is ! result we get equal to f'(x)g'(x)
_8
  #3_4 ,9 ord] 6seek ! correct =mula =!
derivative (a product ( #2 func;ns1 fill
9 ! blanks 9 ! foll[+ ta#3










                                      #a
  ,',key 6items 9 ! foll[+ ta#
#a k(x) .k f(x)g(x)
#b f'(a)
#c g'(a)
#d k'(a)
#e #3x+1
#f #4x+1
#g #5x+1
#h #2x+1
#i #4x+2
#aj x^2"+3x+1
#aa x^2"-2,'
 











                                      #b

-----------------------------------------------
f(x)  g(x)  #a   ;a   f(a)  g(a)  #b   #c   #d
-----------------------------------------------
;x    ;x         #0
-----------------------------------------------
#2x   #3x        #0
-----------------------------------------------
#2x   #e         #0
-----------------------------------------------
#2x   #f         #0
-----------------------------------------------
#3x   x+5        #0
-----------------------------------------------
x+1   #4x        #0
-----------------------------------------------
#g    #4x        #0
-----------------------------------------------
x+2   #7x        #0
-----------------------------------------------
x+1   x+1        #0
-----------------------------------------------
x+2   x+1        #0
-----------------------------------------------
#h    x+1        #0
-----------------------------------------------
#e    #i         #0
-----------------------------------------------
#aj   #aa        #1
-----------------------------------------------
                                      #d
  ,n[ 3jecture a =mula =! derivative
k'(a) 9 t]ms ( f(a), g(a), f'(a), &
g'(a)_4 ,try m func;ns & m po9ts if a
patt]n does n develop1 or sit back &
?9k4 ,! patt]n ( z]os 9 ! f(a) & g(a)
columns is "! 6help y ?9k4
  ,a m di6icult *all5ge3 ,c y ?9k (a way
6e/abli% ! tru? ( yr 3jecture 9 g5]al8
  #4_4 ,6get a =mula =! derivative (a
quoti5t ( #2 func;ns1 let's />t \ easy
0look+ at reciprocals4 ,t is1 we'll let
r(x) .k #1_/g(x), & we'll try 6-pute
r'(x)_4
  ,a natural 3jecture wd 2 t
r'(x) .k #1_/g'(x), i4e41 t
?d_/dx#(1_/g(x)) .k #1_/g'(x)_4 ,try "o
or two func;ns at "o or two po9ts 6te/ ?
3jecture4 ,l9e> func;ns1 or ev5 3/ant
func;ns1 %d su6ice4
  #5_4 ,9 ord] 6seek ! correct =mula =!
derivative ( r(x) .k #1_/g(x), fill 9 !
blanks 9 ! foll[+ ta#4 ,? is a bit m
dem&+ ?an ! previ\s ta#1 s9ce y c't yet
-pute ! derivative r'(a) analytic,y4
                                      #e
,9/1d1 y'll ne$ ei 6zoom 9 6! graph )
,maple & 6try 6e/imate ! slope1 or
6e/imate
  "lim%h $o #0] ?r(a+h)-r(a)_/h#
num]ic,y 0us+ values ( ;h close to #0_4
  ,',key 6items 9 ! foll[+ ta#
#a r(x) .k #1_/g(x),'
 
















                                      #f

---------------------------------------
g(x)       #a   ;a   g(a)  g'(a)  r'(a)
---------------------------------------
;x              #1
---------------------------------------
#2x-1           #1
---------------------------------------
#3x-2           #1
---------------------------------------
#4x-3           #1
---------------------------------------
x+1             #1
---------------------------------------
#2x             #1
---------------------------------------
#3x-1           #1
---------------------------------------
#4x-2           #1
---------------------------------------
x-2             #1
---------------------------------------
#2x-1           #1
---------------------------------------
#7x-3           #1
---------------------------------------
x^2"+3x+1       #0
---------------------------------------
                                      #g
  ,n[ 3jecture a =mula =! derivative
r'(x) 9 t]ms ( g(x) & g'(x)_4 ,try m
func;ns & m po9ts if a patt]n does n
develop1 or sit back & ?9k4 ,! patt]n (
values 9 ! g(a) column is "! 6help y
?9k4
  ,a m di6icult *all5ge3 ,c y ?9k (a way
6e/abli% ! tru? ( yr 3jecture 9 g5]al8
  #6_4 ,-b9e yr =mulas =! derivatives (
products &( reciprocals 6get a =mula =!
derivative (! quoti5t ( #2 func;ns4 ,a
h9t3
  ?f(x)_/g(x)# .k f(x)*?1_/g(x)#
is a product ( #2 func;ns1 "o ( : is a
reciprocal4









                                      #h