,derivatives

    ,review (! ,slope ,pro#m
  ,rememb] f ,limits (,i) ! pro#m t />t$
u \ 9 9v5t+ limits4 ,we want$ 6-pute !
slope (! tang5t l9e 6a curve at a giv5
po9t4 ,9 "picul>1 we ask$ ab ! slope (!
curve y .k x^2 at ! po9t (1, 1)_4 ,\r
approa* 6? 0 6approximate ! tang5t l9e
6! curve 0us+ a secant l9e1 a l9e cutt+
! curve n j at "o po9t b at #2 close tgr
po9ts1 z pictur$ 9 ,limits1 ,figure #1_4
,! second po9t at : ! secant l9e meets !
curve is (t5 writt5 z (x+.,dx, (x+.,d)^2
"), s9ce ? descrip;n hi<li<ts ! fact t
we >e try+ 6pick a po9t a small 4t.e
.,dx away f #1_4 ,! algebra looks a ll
n1t]1 ?\<1 if we use ! simpl] ;h 9/1d (
.,dx, & write ! second po9t z (x+h,
f(x+h))_4 ,! slope (! secant l9e is !n
  ?(1+h)^2"-1_/(1+h)-1#_4
  ,! secant l9e w get clos] & clos] 6!
tang5t l9e z h $o #0, : l1ves u say+ t !
slope (! tang5t l9e %d 2
                                      #j
  "lim%h $o #0] ?(1+h)^2"-1_/(1+h)-1#
    .k "lim%h $o #0] ?((1+h)-1)((1+h)+1)
      _/(1+h)-1#
    .k "lim%h $o #0] ?h(2+h)_/.,dx#
    .k "lim%h $o #0] (2+h) .k #2_4
,? agre$ ) :at num]ical exp]i;t,n &
graph+ _h alr l$ u 63jecture4

    ,def9i;n (! derivative4
  ,n[ let's d ! same ?+ = an >bitr>y
func;n ;f at an >bitr>y po9t ;x_4
,6-pute ! slope (! tang5t l9e to
y .k f(x) at ! "picul> po9t (x, f(x)),
"o takes a second po9t (x+h, f(x+h)) on
! curve1 & "o -putes ! slope (! secant
l9e jo9+ ^! #2 po9ts4 ,! result is
  ?f(x+h)-f(x)_/(x+h)-x#
    .k ?f(x+h)-f(x)_/h#_4
,! slope (! tang5t l9e at (x, f(x)) %d 2
! limit (! slope (! secant l9es z
h $o #0_4 ,? slope is call$ !
.derivative ( ;f at ! po9t x, & x is
writt5
  f'(x)
                                      #a
    .k "lim%h $o #0] ?f(x+h)-f(x)_/h#_4
  ,? :ole process is %[n pictori,y 9
,figure #1_4
  ,! slope (! solid tang5t l9e 9 ,figure
#1 is f'(x)_4 ,! slope (! da%$ secant
l9e is
  ?.,dy_/.,dx# .k ?f(x+h)-f(x)_/h#_4
,if we imag9e ! ;h %r9k+ t[>d #0, s t
x+h $o x, ! secant l9e w sw+ up clos] &
clos] 6! tang5t l9e1 until _! slopes
ne>ly co9cide4 ,! limit (! slopes (!
secant l9es w 2 ! slope (! tang5t l9e3
  f'(x)
    .k "lim%h $o #0] ?f(x+h)-f(x)_/h#_4

    ,a ,not,nal ,9t]lude4
  ,"! >e a _m alt]native ways 6write !
def9i;n (! derivative1 "s simpl] 6look
at ?an o!rs1 b all say+ ! same ?+4 ,z
m5;n$ abv1 "s p don't l us+ ;h z ! 4t.e
2t ! two po9ts4 ,!y pref] .,dx, : looks
obvi\sly l a 4t.e 2t two values ( ;x_4
,= ^! p1 ! two po9ts on ! curve wd 2
writt5 (x, f(x)) & (x+.,dx, f(x+.,dx))_4
                                      #b
,! derivative wd !n 2
  f'(x) .k "lim%.,dx $o #0] ?f(x+.,dx)
    -f(x)_/.,dx#_4
,o!r p don't l ! sum 9 f(x+h) or
f(x+.,dx)_4 ,^! p all ! two po9ts 9 !
figure (a, f(a)) & (x, f(x)), & !y set \
6-pute ! slope f'(a) ( y .k f(x) at !
po9t (a, f(a))_4 ,!y "!=e label !
picture z %[n 9 ,figure #2_4
  ,n[ ! slope (! solid tang5t l9e 9 is
f'(a)_4 ,! slope (! da%$ secant l9e is
  ?.,dy_/.,dx# .k ?f(x)-f(a)_/x-a#_4
,! secant l9e w sw+ up clos] & clos] 6!
tang5t l9e z ;x moves clos] & clos] to
;a_2 s n[ we h 6take a limit z x $o a_4
,! limit (! slopes (! secant l9es w 2 !
slope (! tang5t l9e3
  f'(a)
    .k "lim%x $o a] ?f(x)-f(a)_/x-a#_4
  ,all ^! =mulas >e say+ exactly ! same
?+2 ! only di6];e is ! "ns (! po9ts4
,use :i*"e "o makes y happie/4 ,x's wor?
rememb]+ !y all exi/1 ?\<1 bo? s t y c
talk 6o!r p ) di6]5t not,nal pref];es &
                                      #c
2c "! >e "ts :5 "o ( ^! *oices results 9
simpl] >i?metic ?an ano!r4
  ,! derivative is al n alw writt5 z
f'(x)_4 ,if y .k f(x), !n ! derivative (
;f at ;x c 2 writt5
  f'(x) .k y' .k ?df_/dx# .k ?dy_/dx#
    .k ?d_/dx#f(x) .k ,d;x"f
    .k ,d;x"(y),
plus v>iants on all ^!4 ,! derivative at
! "picul> po9t y .k #2 wd !n 2
      ,'rule.(-0.11in.).(#0.01in.).(#0.3
in.),'
  f'(2) .k y'(2) .k (?dy_/dx#)(2)
    .k ?df_/dx#;x ;.k #2
etc4 ,ag1 all ^! m1n exactly ! same ?+3
! slope (! tang5t l9e to y .k f(x) at a
g5]ic po9t (x, f(x)) &! slope (! tang5t
l9e 6! curve y .k f(x) at ! "picul> po9t
(2, f(2))_4
  ,! not,ns t look l df_/dx >e "picul>ly
9t]e/+ hi/oric,y4 ,! two ma9 9v5tors (
calculus 7 ,sir ,isaac ,newton 9 ,5gl& &
,gottfri$ ,wilhelm1 ,freih]r von
,leibniz 9 ,g]_m4 ,newton us$ not,n sort
                                      #d
( l y' =! derivative (actu,y1 he us$
"y<*])1 :ile ,leibniz us$ ?dy_/dx#_4
,leibniz _h no no;n ( limits (nor _h
,newton)1 s 8 ?"\ 0 ?3 ! slope (! secant
l9e 2t ! po9t (x, f(x)) &! po9t (x+.,dx,
f(x+.,dx)) is
  ?f(x+.,dx)-f(x)_/.,dx#
    .k ?.,df_/.,dx# .k ?.,dy_/.,dx#_4
,leibniz imag9$ -put+ ! slope (! tang5t
l9e 0tak+ #2 po9ts t 7 only an
9f9itesimal 4t.e a"p---a 4t.e too small
6be writt5 9 decimal =m1 b / n z]o4 ,he
denot$ ? t9y---& imag9>y---_4t.e z dx_4
,! correspond+ *ange 9 ! hei<t (! curves
!n dy .k df_2 &! slope (! secant l9e1 :
0 re,y equal 6! tang5t l9e1 0
?dy_/dx# .k ?df_/dx#_4 ,? idea is re,y r
3fus$1 & 3fus+1 b x gives rise 6not,n t
captures an important 9si<t1 t !
derivative is alm equal 6! slope ?.,dy
_/.,dx# (a secant l9e1 = v1 v small
values ( .,dy & .,dx_4 ,x's al a v nice
%orth&---_6take a limit1 j *ange
alphabets1 replac+ ! ,greek .,d )a ,lat9
                                      #e
;d_4 ,9 3d5s$1 s*ematic =m1
  "lim%.,dx $o #0] ?.,df_/.,dx#
    .k ?df_/dx#_4

    ,derivatives z ,rates ( ,*ange4
  ,"h's ano!r way 6?9k ab derivatives4
,suppose f(t) 7 ! posi;n ( an object at
"t ;t_4 ,!n f(t+.,dt)-f(t) wd 2 ! 4t.e
travell$ 2t "t ;t & "t t+.,dt_4 ,!
di6];e quoti5t
  ?f(t+.,dt)-f(t)_/.,dt#
wd 2 ! av]age veloc;y 2t ^! two "ts1 i4
e41 ! av]age rate ( *ange (! posi;n ;f
2t "ts ;t & t+.,dt_4 ,! derivative
  f'(t) .k "lim%.,dt $o #0] ?f(t+.,dt)
    -f(t)_/.,dt#
is ! limit ( t av]age veloc;y ov] v
small 9t]vals 9clud+ "t ;t_4 ,9 o!r ^ws1
x is ! 9/antane\s veloc;y (! object
exactly at "t 1 ! 9/antane\s rate (
*ange ( ;f at "t ;t_4 ,! limit 3cept
lets u make s5se (! veloc;y or rate (
*ange (a quant;y n ov] an 9t]val---!
av]age veloc;y or rate ( *ange---b at an
                                      #f
9/ant 9 "t---! 9/antane\s veloc;y or
rate ( *ange4
  ,! r1son we c>e s m* ab ! slope pro#m1
&! r1son calculus is s important a "p (!
ma!matical sci5ti/'s >ma;t>ium is t 9
-put+ slopes1 we >e re,y -put+
9/antane\s rates ( *ange4 ,\r abil;y 6d
? is critical 6\r abil;y 6model any
quant;y 9 ! univ]se t is a# 6*ange4

    ,"s f/ -put,ns4
  ,on ! second "d ( class1 we made a
3jecture1 t ! slope (! tang5t l9e 6!
curve y .k x^2 at ! po9t (x, x^2") wd 2
#2x_4 ,we gave qualitative r1sons 6?9k ?
0 true4 ,we "k x is exactly true at
x .k #0_4 ,! calcul,n at ! top ( ? h&\t
%[s t x is exactly true at x .k #1_4
,0symmetry1 x m/ 2 true at x .k -#1 z
well4 ,n[ let's d x 9 g5]al4 ,0! def9i;n
(! derivative1 ! slope (! tang5t l9e to
y .k x^2 at ! po9t (x, f(x)) is
  f'(x)
    .k "lim%h $o #0] ?f(x+h)-f(x)_/h#
                                      #g
    .k "lim%h $o #0] ?(x+h)^2"-x^2"_/h#
    .k "lim%h $o #0] ?(x^2"+2xh+h^2")
      -x^2"_/h#
    .k "lim%h $o #0] ?2xh+h^2"_/h#
    .k "lim%h $o #0] (2x+h) .k #2x_4
,! 3jecture we made at ! 2g9n+ (! t]m t
at l1/ =! func;n y .k x^2, ! >ea & slope
pro#ms >e 9v]ses ( "o ano!r1 is "!=e
exactly correct2 & !calcul,n is n ev5 v
h>d4 ,we've le>n$ a lot6
  ,x's wor? plott+ ! func;n f(x) .k x^2
& xs derivative f'(x) .k #2x side 0side
9 ord] 6?9k ab ! geometric rel,n 2t !m4
  ,! derivative ( ;f at ! po9t ;x is !
slope (! tang5t l9e to ;f at ! po9t
(x, f(x))_4 ,t is1 at any giv5 po9t ;x
on ! ;x-axis1 ! hei<t (! curve 9 ,figure
#4 %d 2 ! slope (! curve 9 ,figure #3_4
,try 6e/imate ! slope 9 ,figure #3 at a
few po9ts & 63v9ce yrf t ! curve 9
,figure #4 is re,y giv+ y ^! slopes4
  ,let's try #2 m examples 2f we pause &
look >.d4 ,let g(x) .k x^3, & let u
-pute ! slope (! tang5t l9e to y .k g(x)
                                      #h
at ! po9t (x, g(x))_4 ,0! def9i;n (!
derivative1 ? %d 2
  g'(x)
    .k "lim%h $o #0] ?g(x+h)-g(x)_/h#
    .k "lim%h $o #0] ?(x+h)^3"-x^3"_/h#
    .k "lim%h $o #0] ?(x^2"+3x^2"h+3xh^2
      "+h^3")-x^3"_/h#
    .k "lim%h $o #0] ?3x^2"h+3xh^2"+h^3"
      _/h#
    .k "lim%h $o #0] (3x^2"+3xh+h^2")
      .k #3x^2_4
,! graphs ( y .k g(x) & y .k g'(x) >e
%[n 9 ,figures #5 & #6, resp4 ,ag1 !
hei<t (! curve 9 ,figure #6 %d 2 ! slope
(! curve 9 ,figure #5_4
  ,f9,y1 :at if k(x) .k ?1_/x#_8 ,!n !
slope (! tang5t l9e to y .k k(x) at !
po9t (x, ?1_/x#) %d 2
  k'(x)
    .k "lim%h $o #0] ?k(x+h)-k(x)_/h#
    .k "lim%h $o #0] ,??1_/x+h#-?1_/x#
      ,_/h,#
    .k "lim%h $o #0] ?1_/h#@(?1_/x+h#-?1
      _/x#@)
                                      #i
    .k "lim%h $o #0] ?1_/h#@(?x_/x(x+h)#
      -?x+h_/x(x+h)#@)
    .k "lim%h $o #0] ?1_/h#@(-?h_/x(x
      +h)#@)
    .k "lim%h $o #0] @(-?1_/x(x+h)#@)
      .k -?1_/x^2"#_4
,! graphs ( y .k k(x) & y .k k'(x) >e
%[n 9 ,figures #7 & #8, resp4 ,! hei<t
(! curve 9 ,figure #8 %d 2 ! slope (!
curve 9 ,figure #7_4

    ,r1l;y ,*eck4
  ,if y .k mx+b is a /rai<t l9e1 we n[ h
#2 me?ods = -put+ ! slope ( ;y at ! po9t
(x, mx+b)_4 ,we cd rememb] f s*ool t !
slope is alw m, or we cd -pute !
derivative4 ,wd we get ! same answ]8
,let's f9d4 ,6make ! calcul,n m 9t]e/+1
,i'll make "s di6]5t not,nal *oices ?an
we h previ\sly4
  ,! slope (! curve y .k mx+b at ! po9t
(x, mx+b) is suppos$ 6be
  ?.,dy_/.,dx#
    .k "lim%.,dx $o #0] ?@(m(x+.,dx)+b@)
                                     #aj
      -@(mx+b@)_/.,dx#
    .k "lim%.,dx $o #0] ?mx+m.,dx+b-mx-b
      _/.,dx#
    .k "lim%.,dx $o #0] ?m.,dx_/.,dx#
      .k "lim%.,dx $o #0] m .k m_4
,? is exactly :at !y tau<t u 9 s*ool---&
:at we us$ z \r />t+ po9t = ? :ole !ory4
,s \r new me?od ( -put+ slopes doesn't
=ce u 6*ange ! m1n+ ( slopes =! only
curve = : we cd -pute ! slope )\t
calculus1 ! /rai<t l9e4

    ,typical ,pro#ms4

,pro#m #1_4
  ,-pute ! slope (! tang5t l9e 6! curve
y .k x^2"-3x+2 at ! po9t (2, 0)_4
  ,"o way 6d ? pro#m wd 2 9 ! 3crete
/yle we us$ = \r f/ slope pro#m4 ,!
secant l9e jo9+ ! po9t at (2, 0) & ano!r
po9t at (2+.,dx,
(2+.,dx)^2"-3(2+.,dx)+2) has slope
  ?@((2+.,dx)^2"-3(2+.,dx)+2@)-@(2^2"
    -3(2)+2@)_/.,dx#
                                     #aa
    .k ?@((2+.,dx)^2"-3(2+.,dx)+2@)
    -@(0@)_/.,dx#_4
,! tang5t l9e at (2, 0) wd "!=e h slope
  y'
    .k "lim%.,dx $o #0] ?@((2+.,dx)^2"-
      3(2+.,dx)+2@)-@(0@)_/.,dx#
    .k "lim%.,dx $o #0] ?4+4.,dx
      +(.,dx)^2"-6-3.,dx+2_/.,dx#
    .k "lim%.,dx $o #0] ?.,dx+(.,dx)^2"
      _/.,dx#
    .k "lim%.,dx $o #0] (1+.,dx) .k #1_4
  ,an alt]native approa* wd 2 6-pute !
derivative at an >bitr>y po9t x, & !n
6plug 9 x .k #2_4 ,9 ? me?od1 we wd f/
-pute
  f'(x)
    .k "lim%h $o #0] ?@((x+h)^2"-3(x+h)+
      2@)-@(x^2"-3x+2@)_/h#
    .k "lim%h $o #0] ?x^2"+2xh+h^2"-3x-
      3h+2-x^2"+3x-2_/h#
    .k "lim%h $o #0] ?2xh+h^2"-3h_/h#
    .k "lim%h $o #0] (2x+h-3) .k #2x-3_4
,? is ! slope (! tang5t l9e at ! po9t
(x, x^2"-3x+2)_4 ,! slope (! tang5t l9e
                                     #ab
at (2, 0) is "!=e f'(2) .k #2(2)-3 .k #1
, ! same answ] we got 2f4

,pro#m #2_4
  ,-pute ! equ,n (! tang5t l9e 6! curve
y .k x^2"-3x+2 at ! po9t (2, 0)_4
  ,! f/ "p ( ? pro#m wd 2 6d ,pro#m #1,
-put+ ! slope (! tang5t l9e4 ,once we've
d"o t1 ! pro#m is easy4 ,! tang5t l9e
has slope #1, & x goes "? ! po9t (2, 0)
_2 s xs equ,n m/ 2 (y-0) .k #1(x-2), i4
e41 y .k x-2_4 ,z a *eck1 ! curve & ?
l9e >e plott$ tgr 9 ,figure #9_4 ,!y
look pretty tang5t4

    ,func;ns )\t derivatives4
  ,we've "w$ \ n[ h[ 6-pute !
derivatives ( q a lot ( func;ns1 & we're
ab 6/>t look+ = g5]al rules t w let u
di6]5tiate practic,y e func;n4 ,b 2f we
d t1 we "\ 6pause & cl>ify sli<tly \r
idea ( derivatives 0obs]v+ t n e func;n
has a derivative at e po9t4 ,f/ ( all1
only a func;n t is 3t9u\s at x .k a c h
                                     #ac
a derivative at x .k a_4 ,? seems
r1sona# 53 surely any func;n t c't ev5 2
drawn at x .k a )\t pick+ up "os p5cil
isn't 2 smoo? 5 6h a tang5t l9e at ;a_4
,"h's a cl"e >gu;t t actu,y proves ?
fact analytic,y us+ \r def9i;ns (
derivative &( 3t9u;y3 ,suppose
  f'(a) .k "lim%x $o a] ?f(x)-f(a)_/x-a#
exi/s4 ,!n
  "lim%x $o a] (f(x)-f(a))
    .k "lim%x $o a] (?f(x)-f(a)_/x-a#*(x
      -a))
    .k ("lim%x $o a] ?f(x)-f(a)_/x-a#)(
      "lim%x $o a] (x-a))
    .k f'(a)*0 .k #0,
: m1ns t
  "lim%x $o a] f(x)
    .k "lim%x $o a] ((f(x)-f(a))+f(a))
    .k "lim%x $o a] (f(x)-f(a))+
      "lim%x $o a] f(a)
    .k #0+f(a) .k f(a),
: m1ns t ;f is 3t9u\s at ;a_4
  ,does e func;n t is 3t9u\s at ;a & h a
derivative at ;a_8 ,no4 ,3sid]1 = 9/.e1
                                     #ad
f(x) .k \x\ at a .k #0_4 ,! derivative
f'(0), if x exi/$1 wd 2
  "lim%h $o #0] ?f(0+h)-f(0)_/h#
    .k "lim%h $o #0] ?\0+h\-0_/h#
    .k "lim%h $o #0] ?\h\_/h#_4
  ,if h .1 #0, !n \h\ .k h, :ile if
h "k #0, !n \h\ .k -h_4 (,?9k ab \2\ &
\-2\ if y don't see ?4) ,?us1
  "lim%h $o #0^+]  ?\h\_/h#
    .k "lim%h $o #0^+]  #1 .k #1,
b
  "lim%h $o #0^-]  ?\h\_/h#
    .k "lim%h $o #0^-]  (-1) .k -#1_4
,s9ce ! left & "r h& limits di6]1
"lim%h $o #0] ?\h\_/h# does n exi/1 &
y .k \x\ does n h a derivative at
x .k #0_4 ,? seems r1sona# if y ?9k (!
graph ( \x\ (,figure #10)_4  ,:at cd "o
m1n 0! slope (! tang5t l9e 6? func;n at
x .k #0, ": ! graph -es 6a %>p po9t8

    ,basic rules = -put+ derivatives4


                                     #ae
,p[]s ( ;x_4
  ,s f>1 we "k t
  ?d_/dx#(x) .k #1
  ?d_/dx#(x^2") .k #2x
  ?d_/dx#(x^3") .k #3x^2_4
,is "! any patt]n "h8 ,h[ ab
  ?d_/dx#(x^n") .k nx^n-1_4
  ,6prove t ? 3jectur$ =mula "ws = any
positive 9teg] value ( ;n, we wd h 6let
f(x) .k x^n & 6evaluate
  f'(a)
    .k "lim%h $o #0] ?(a+h)^n"-a^n"_/h#
    .k "lim%x $o a] ?x^n"-a^n"_/x-a#_4
  ,! second =m abv seems 6be ! easi] "o
6"w )1 s9ce we c imag9e j do+ l;g divi.n
6divide x-a 9to x^n"-a^n (try it6) 6get
  x^n"-a^n
    .k (x-a)(x^n-1"+ax^n-2"+a^2"x^n-3"
    + ''' +a^n-2"x+a^n-1")_4
,? m1ns t
  "lim%x $o a] ?x^n"-a^n"_/x-a#
    .k "lim%x $o a] (x^n-1"+ax^n-2"+a^2
      "x^n-3"+ ''' +a^n-2"x+a^n-1")
    .k (a^n-1"+aa^n-2"+a^2"a^n-3"+ '''
                                     #af
      +a^n-2"a+a^n-1")
    .k na^n-1,
j ! =mula we 7 hop+ =4 ,if y want an
9t]e/+ algebraic *all5ge1 y mi<t 5joy
see+ t y get ! same result :5 y -pute
  f'(a) .k "lim%h $o #0] ?(a+h)^n"
    -a^n"_/h#_4
,rememb]+ at l1/ r\<ly :at ! ,binomial
,!orem says w 2 a help "h4
  ,! =mula t (x^n")' .k nx^n-1 "ws m
g5],y ?an j = positive 9teg] values (
;n, ?\<4 ,= 9/.e1 we h alr se5 t
  ?d_/dx#(x^-1") .k ?d_/dx#(?1_/x#)
    .k -?1_/x^2"# .k (-1)x^-2,
: is ! same =mula ) n .k -#1_4
  ,we cd al try ! case n .k ?1_/2#, :
m1ns we h 6-pute ! derivative (
f(x) .k x^1_/2 .k >x]_4 ,we get
  f'(x)
    .k "lim%h $o #0] ?>x+h]->x]_/h#
    .k "lim%h $o #0] (?>x+h]->x]_/h#*?>x
      +h]+>x]_/>x+h]+>x]#)
    .k "lim%h $o #0] ?(x+h)-x_/h(>x+h]
      +>x])#
                                     #ag
    .k "lim%h $o #0] ?h_/h(>x+h]+>x])#
    .k "lim%h $o #0] ?1_/>x+h]+>x]#
    .k ?1_/2>x]# .k ?1_/2#x^-1_/2,
! result (! =mula nx^n-1 ) n .k ?1_/2#_4
  ,we'll -e back lat] 6! =mula
  ?d_/dx#(x^n") .k nx^n-1
lat]1 & see if we c prove xs valid;y =
all r1l ;n_4

,3/ant multiples4
  ,n[ t we "k h[ 6di6]5tiate at l1/ e
positive 9teg] p[] ( ;x, let's see ab
-b9+ ^! p[]s 6make polynomials4
  ,! f/ ?+ we ne$ 6be a# 6d is 6multiply
p[]s ( ;x 03/ants 6get func;ns l #3x^5_4
,"h ! !orem is simple3 ,if ;c is a
3/ant1 !n
  (cf(x))' .k c*f'(x)_4
,6prove ?1 we -pute
  ?d_/dx#(cf(x))
    .k "lim%h $o #0] ?cf(x+h)-cf(x)_/h#
    .k "lim%h $o #0] (c*?f(x+h)-f(x)
      _/h#)
    .k ("lim%h $o #0] c)("lim%h $o #0]
                                     #ah
      ?f(x+h)-f(x)_/h#)
    .k c*f'(x)_4
,we "!=e "k ?+s l
  ?d_/dx#(3x^5") .k #3?d_/dx#(x^5")
    .k #3*5x^4 .k #15x^4_4

,sums & ,di6];es4
  ,6get polynomials1 we n[ h 6add up
t]ms t look l #3x^5_4 ,6take !
derivative (a sum ( func;ns is al
simple3
  (f(x)+g(x))' .k f'(x)+g'(x)_4
,ag1 ! pro( ( ? is a simple -put,n3
  ?d_/dx#(f(x)+g(x))
    .k "lim%h $o #0] ?(f(x+h)+g(x+h))
      -(f(x)+g(x))_/h#
    .k "lim%h $o #0] ?(f(x+h)-f(x))+(g(x
      +h)-g(x))_/h#
    .k "lim%h $o #0] (?f(x+h)-f(x)_/h#+
      ?g(x+h)-g(x)_/h#)
    .k ("lim%h $o #0] ?f(x+h)-f(x)_/h#)
      +("lim%h $o #0] ?g(x+h)-g(x)_/h#)
    .k f'(x)+g'(x)_4

                                     #ai
  ,0exactly ! same >gu;t1 ! derivative
(! di6];e ( two func;ns is
  (f(x)-g(x))' .k f'(x)-g'(x)_4
  ,0-b9+ ! results we h s f>1 we c -pute
! derivative ( any polynomial4 ,= 9/.e1
  ?d_/dx#(x^17"-11x^7"+6x^2"+13)
    .k (x^17")'-(11x^7")'+(6x^2")'+(13)'
    .k (x^17")'-11(x^7")'+6(x^2")'+(13)'
    .k #17x^16"-77x^6"+12x+0_4
(,rememb] t a 3/ant func;n is a
horizontal /rai<t l9e ) slope #0, s t xs
derivative %d 2 #0_4 ,y c al get ? "r \
(! def9i;n ( derivative appli$ 6a 3/ant
func;n_4)

    ,preview4
  ,we c n[ -pute ! derivatives ( an
enorm\s collec;n ( func;ns4 ,>e we d"o8
,no4 ,"! >e / lots ( func;ns we c't
di6]5tiate4 ,^! 9clude
  ,trig func;ns1 l sin x_4
  ,products ( func;ns1 l x^2"*sin x_4
  ,quoti5ts ( func;ns1 l
?x^2"+x-1_/5x+2#_4
                                     #bj
  ,p[]s ( func;ns1 l (4x^2"+x+11)^100,
or 9 g5]al1 func;ns ( func;ns1 l
sin (x^2")_4
  ,"!'s al ! matt] ( non-positive-9teg]
p[]s ( ;x, l x^-5 or x^22_/7_4
  ,s we / ne$ "s m rules1 b we h made
enorm\s h1dway 9 v %ort ord]4

    ,derivatives ( sin x & cos x_4
  ,if we 7 6guess :at ! derivatives (!
s9e & cos9e func;ns 7 l1 :at wd we say8
,! graph ( y .k sin x is %[n 9 ,figure
#11_4
  ,! derivative ( y .k sin x has 6be
"s?+ l ! func;n graph$ 9 ,figure #12_4
  ,! func;n 9 ,figure #12 looks at l1/
sup]fici,y l ! func;n y .k cos x, s a
r1sona# 3jecture wd 2 t
  ?d_/dx#(sin x) .k cos x_4
  ,simil>ly1 ! graph ( y .k cos x &&
approxim,n 6xs derivative >e %[n 9
,figures #13 & #14, resp4
  ,x wd appe> t p]h
  ?d_/dx#(cos x) .k -sin x_4
                                     #ba
  ,c we prove ^! claims8 ,x turns \ (
maybe surpris+ly) t ! answ] is yes4
,6get ! derivative ( sin x, we say1 if
f(x) .k sin x, !n
  f'(x)
    .k "lim%h $o #0] ?sin (x+h)-sin x
      _/h#
    .k "lim%h $o #0] ?sin xcos h+cos x
      sin h-sin x_/h#
    .k "lim%h $o #0] ?sin x(cos h-1)+
      cos xsin h_/h#
    .k ("lim%h $o #0] ?sin x(cos h-1)
      _/h#)+("lim%h $o #0] ?cos xsin h
      _/h#)
    .k ("lim%h $o #0] sin x)(
      "lim%h $o #0] ?(cos h-1)_/h#)+(
      "lim%h $o #0] cos x)("lim%h $o #0]
      ?sin h_/h#)
    .k ("lim%h $o #0] sin x)*0+(
      "lim%h $o #0] cos x)*1
    .k (sin x)*0+(cos x)*1 .k cos x_4
,9 l9e #2 ( ? calcul,n1 we h us$ ! sum
=mula =! s9e func;n4 ,9 l9e #6, we use
#2 (! limits we 3jectur$ 9 ! lab & lat]
                                     #bb
prov$ 9 class4
  ,6-pute ! derivative ( cos x, we cd
try a simil> calcul,n us+ ! sum =mula =!
cos9e4 ,we cd al obs]ve t ! graph (!
cos9e func;n is ! graph (! s9e func;n
mov$ left 0a 4t.e .p_/2_4
  cos x .k sin (x+?.p_/2#)_4
,! derivative (! cos9e func;n %d "!=e 2
! derivative (! s9e func;n mov$ left 0a
4t.e .p_/2_4 ,t is1
  ?d_/dx#(cos x) .k"
    .k ?d_/dx#(sin (x+?.p_/2#))
    .k cos (x+?.p_/2#)_4
,look+ at ! unit circle or us+ "s ( \r
e>li] trig id5tities %[s t
cos (x+.p_/2) .k -sin x, s t
  ?d_/dx#(cos x) .k cos (x+?.p_/2#)
    .k -sin x_4

    ,! ,product ,rule4
  ,! ,product ,rule says
  ?d_/dx#(f(x)g(x))
    .k f'(x)g(x)+f(x)g'(x)_4
or1 9 %orth&1
                                     #bc
  (fg)' .k f'g+fg'_4
,! pro( is a 0n[ /&>d trick ( a4+ &
subtract+ ! same ?+4 ,m* ( ma!matics
3si/s ( exactly ?3 a4+ #0 & multiply+ by
#1 9 cl"e ways4
  ?d_/dx#(f(x)g(x))
    .k "lim%h $o #0] ?f(x+h)g(x+h)
      -f(x)g(x)_/h#
    .k "lim%h $o #0] ?f(x+h)g(x+h)
      -f(x)g(x+h)+f(x)g(x+h)-f(x)g(x)
      _/h#
    .k "lim%h $o #0] @(?f(x+h)g(x+h)
      -f(x)g(x+h)_/h#+?f(x)g(x+h)
      -f(x)g(x)_/h#@)
    .k "lim%h $o #0] @(?f(x+h)g(x+h)
      -f(x)g(x+h)_/h#@)+"lim%h $o #0] @(
      ?f(x)g(x+h)-f(x)g(x)_/h#@)
    .k ("lim%h $o #0] ?f(x+h)-f(x)_/h#)(
      "lim%x $o #0] g(x+h))+(
      "lim%h $o #0] f(x))("lim%h $o #0]
      ?g(x+h)-g(x)_/h#)
    .k f'(x)g(x)+f(x)g'(x)_4
  ,x may 9iti,y seem surpris+ t
(fg)' /.k f'g'_4 ,"h's a picture t may
                                     #bd
cl>ify :at's happ5e+ 9 ! ,product ,rule4
,suppose f(t) repres5ts ! l5g? (a
rectangle & g(t) repres5ts xs wid?4 ,bo?
^! quantities >e *ang+ ) "t---!
rectangle is gr[+4 ,! func;n f(t)g(t) !n
repres5ts ! >ea (! rectangle2 &
?d_/dt#(f(t)g(t)) is ! rate ( *ange ( ?
>ea4
  ,! picture 9 ,figure #15 %[s !
rectangle at "t ;t (solid) al;g )! gr[?
(! rectangle 2t "t ;t & "t t+.,dt (da%$
)_4
  ,2t "t ;t & "t t+.,dt, ! >ea (! solid
rectangle 9cr1ses 2c (! a4i;n ( ?ree
small] rectangles1 ,a, ,b, & ;,c_4 ,!
rectangle ;,a has l5g? f(t) & wid? ab
g'(t).,dt_4 ,rectangle ;,b has l5g? g(t)
& wid? ab f'(t).,dt_4 ,! heigt & wid? (
rectangle ;,c >e ab f'(t).,dt &
g'(t).,dt_4 ,?us1 ! 9cr1se 9 >ea (!
reactangle 2t "t ;t & "t t+.,dt is r\<ly
  f(t)g'(t).,dt+g(t)f'(t).,dt
    +f'(t)g'(t)(.,dt)^2_4
,! rate ( 9cr1se (! >ea is ab
                                     #be
  ?f(t)g'(t).,dt+g(t)f'(t).,dt
    +f'(t)g'(t)(.,dt)^2"_/.,dt#
    .k f(t)g'(t)+g(t)f'(t)
    +f'(t)g'(t).,dt_4
,z .,dt $o #0, ? rate ( *ange approa*es
  ?d_/dt#(f(t)g(t))
    .k f(t)g'(t)+g(t)f'(t),
: is j :at ! ,product ,rule claim$4 ,!
two t]ms 9 ! ,product ,rule "!=e
correspond 6! rates ( 9cr1se ( >ea caus$
0expan.n (! two sides (! rectangle1
resp4

    ,! ,quoti5t ,rule
  ,! rule = di6]5tiat+ reciprocals is
  ?d_/dx#(?1_/g(x)#)
    .k -?g'(x)_/g(x)^2"#_4
,"o c 3v9ce onself ( ? 0j do+ ! only
calcul,n "o cd d3
  ?d_/dx#(?1_/g(x)#)
    .k "lim%h $o #0] ,??1_/g(x+h)#-?1
      _/g(x)#,_/h,#
    .k "lim%h $o #0] ?1_/h#@(?1_/g(x+h)#
      -?1_/g(x)#@)
                                     #bf
    .k "lim%h $o #0] ?1_/h#@(?g(x)-g(x
      +h)_/g(x)g(x+h)#@)
    .k "lim%h $o #0] @(?1_/g(x)g(x+h)#*
      ?g(x)-g(x+h)_/h#@)
    .k ("lim%h $o #0] ?1_/g(x)g(x+h)#)(
      "lim%h $o #0] ?g(x)-g(x+h)_/h#)
    .k ?1_/g(x)^2"#(-g'(x)) .k -?g'(x)
      _/g(x)^2"#_4
  ,! =mula =! derivative ( an >bitr>y
quoti5t c n[ 2 deriv$ 0-b9+ ! ,product &
,reciprocal ,rules3
  ?d_/dx#(?f(x)_/g(x)#)
    .k ?d_/dx#(f(x)*?1_/g(x)#)
    .k f'(x)?1_/g(x)#+f(x)(?d_/dx#(?1
      _/g(x)#))
    .k ?f'(x)_/g(x)#+f(x)(-?g'(x)
      _/g(x)^2"#)
    .k ?g(x)f'(x)-f(x)g'(x)_/g(x)^2"#_4
  ,! ,quoti5t ,rule is h>d] 6rememb] ?an
! ,product ,rule 2c x has t m9us sign4
,y h 6get ! t]ms 9 ! "r ord]4 ,"o solu;n
6? is a bit ( mnemonic do7]el3
6di6]5tiate ?,hi_/,ho#, we say1 8
,ho-de-,hi m9us ,hi-de-,ho1 ,squ>e !
                                     #bg
bottom1 & away we g60 8,de0 "h m1ns
derivative4 ,! rhyme =ces y 6get ! t]ms
9 ! "r ord]2 s9ce ! alt]native is 8
,hi-de-,ho m9us ,ho-de-,hi1 ,squ>e !
bottom1 & we all fry60

    ,o!r ,trig ,func;ns4
  ,an imm use (! quoti5t rule is 6-pute
! derivatives (! o!r trig func;ns4
  ?d_/dx#sec x
    .k ?d_/dx#(?1_/cos x#)
    .k -,?(d_/dx)(cos x),_/cos^2 x,#
    .k ?sin x_/cos^2 x#
    .k ?1_/cos x#*?sin x_/cos x# .k
      sec xtan x_4
  ?d_/dx#tan x
    .k ?d_/dx#(?sin x_/cos x#)
    .k ,?(cos x)((d_/dx)sin x)-(
      sin x)((d_/dx)cos x),_/cos^2 x,#
    .k ?(cos x)(cos x)-(sin x)(-sin x)_/
      cos^2 x#
    .k ?1_/cos^2 x# .k sec^2 x_4
  ,calcul,ns j l ^! give ! derivatives
(! cosecant & cotang5t func;ns4 ,if we
                                     #bh
li/ all ^! results tgr 9 a ta#1 "o c see
"s (! symmetries 9volv$4

------------------
f(x)   f'(x)
------------------
sin x  cos x
------------------
cos x  -sin x
------------------
tan x  sec^2 x
------------------
cot x  -csc^2 x
------------------
sec x  sec xtan x
------------------
csc x  -csc xcot x
------------------

    ,examples4
  ?d_/dx#(xsin x) .k #1sin x+xcos x
  ?d_/dx#(sin^2 x) .k ?d_/dx#(sin x*
    sin x) .k cos xsin x+sin xcos x .k #
    2sin xcos x
                                     #bi
  ?d_/dx#(?x^2"+3x+5_/7x^3"+11#) .k ?(
    7x^3"+11)(2x+3)-(x^2"+3x+5)(21x^2")
    _/(7x^3"+11)^2"#
  ?d_/dx#(?sin xcos x_/x^2"+1#) .k
    ,?(x^2"+1)?d_/dx#(sin xcos x)-(sin x
    cos x)2x,_/(x^2"+1)^2",#
  .k ?(x^2"+1)(cos^2 x-sin^2 x)-(sin x
    cos x)2x_/(x^2"+1)^2"#
  ?d_/dx#(?x_/2x+3#*?tan x_/>x]#) .k ?(
    2x+3)1-x(2)_/(2x+3)^2"#?tan x_/>x]#+
    ?x_/2x+3#,?>x]sec^2 x-(tan x)(?1_/
    2>x]#),_/x,#_4

    ,hi<] ,ord] ,derivatives4
  ,we've be5 />t+ )a func;n f(x) & -put+
xs derivative f'(x)_4 ,b f'(x) is a
func;n 9 xs [n "r4 ,we c "!=e p]fectly
well imag9e tak+ xs derivative 6get yet
ano!r func;n1 f''(x)_4 ,we cd1 if we
want$1 keep go+ & take ! derivative (
f''(x), : is call$ f'''(x) & s on4 ,^!
func;ns >e call$ ! second derivative ( f
, ! ?ird derivative ( f, & s on4
,derivatives hi<] ?an ! ?ird >e norm,y
                                     #cj
writt5 us+ a sup]script 9 p>5!ses
6denote ! ord] (! derivative4 ,?us1 we
write
  f'''(x) .k f^(3)"(x),
  f''''(x) .k f^(4)"(x),
  f'''''(x) .k f^(5)"(x),
& s on4 ,o3a.n,y1 x is al 3v5i5t 6use ?
not,n = l[] ord] derivatives1 usu,y 9
=mulas 9 : we >e summ+ ov] _m
derivatives4 ,we cd !n write
  f''(x) .k f^(2)"(x)
  f'(x) .k f^(1)"(x)
  f(x) .k f^(0)"(x)_4
  ,if we >e us+ ,leibniz,8 not,n =
derivatives1 & if y .k f(x), !n
  f''(x) .k ?d_/dx#(?df_/dx#) .k ?d^2"f
    _/dx^2"# .k ?d^2"y_/dx^2"# .k ?d^2"
    _/dx^2"#f(x),
  f'''(x) .k ?d_/dx#(?d^2"f_/dx^2"#) .k
    ?d^3"f_/dx^3"# .k ?d^3"y_/dx^3"# .k
    ?d^3"_/dx^3"#f(x),
& s on4
  ,= polynomials1 hi<] ord] derivatives
ev5tu,y get simpl] & simpl]4 ,"h's an
                                     #ca
example3
  f(x) .k x^3"-27x^2"+11x+6
  f'(x) .k #3x^2"-54x+11
  f''(x) .k #6x-54
  f'''(x) .k #6
  f^(4)"(x) .k f^(5)"(x) .k f^(6)"(x)
    .k ''' .k #0_4
  ,= o!r func;ns1 ! hi<] ord]
derivatives may get m -plicat$1 n less4
  f(x) .k ?x+1_/3x^2"-1#
  f'(x) .k ?(3x^2"-1)1-(x+1)(6x)_/(3x^2"
    -1)^2"#
  .k -?3x^2"+6x+1_/9x^4"-6x^2"+1#
  f''(x) .k -?(9x^4"-6x^2"+1)(6x+6)-(
    3x^2"+6x+1)(36x^3"-12x)_/(9x^4"-6x^2
    "+1)^2"#
  .k ?18x^3"+54x^2"+18x+6_/27x^3"-27x^2"
    +9x-1#_4
  ,=! s9e & cos9e func;ns1 ! derivatives
cycle3
  f(x) .k sin x
  f'(x) .k cos x
  f''(x) .k -sin x
  f'''(x) .k -cos x
                                     #cb
  f^(4)"(x) .k sin x
  f^(5)"(x) .k cos x_4

    ,a ,physical ,9t]pret,n ( ,hi<]
    ,derivatives4
  ,i once rode an elevator d[n f ! #39?
floor a"p;t ( my fr ,*>lie ,fe6]man1 :o
at ! "t 0 #22, a full professor at
,*icago & at ,pr9ceton1 & "o (! be/
ma!matical analy/s 9 ! _w4 ,i -;t$ t !
elevator 0 v fa/ b v smoo?4 ,*>lie no4$4
8,small ?ird derivative10 he sd4
  ,cd he possibly h be5 "r8 ,c "o feel
?ird derivatives8
  ,69ve/igate ? "q1 suppose y wake up "o
morn+ & f9d yrf 9 an elevator4 ,? wd
raise all mann] ( "qs t >e \tside !
scope ( ? class1 b x wd al raise "qs we
c actu,y a4ress4
  ,let s(t) 2 ! posi;n (! elevator z a
func;n ( "t4 ,9 ,am]ican units1 s(t)
mi<t 2 m1sur$ 9 feet abv ! gr.d4 ,:5 y
wake up 9 ! elevator1 y h no way
6directly exp]i;e ! value ( s(t)_4 ,6be
                                     #cc
on ! f/ floor or ! #50? floor feels !
same4 ,! only way 6det]m9e s(t) is 6r1d
! li<ts or li/5 6! bells %[+ ! floor4
,if y c't see ! li<ts or he> ! bells1 !n
y don't "k ": y >e4
  ,! derivative s'(t) is ! rate ( *ange
(! hei<t (! elevator4 ,we "k t
  s'(t) .k "lim%.,dt $o #0] ?s(t+.,dt)
    -s(t)_/.,dt#_4
,! num]ator1 .,ds, 9 ! di6];e quoti5t is
m1sur$ 9 feet1 &! denom9ator1 .,dt, is
m1sur$ 9 seconds4 ,! rate ( *ange1 s'(t)
, w "!=e h units ( feet p] second4 s'(t)
is ! veloc;y at : ! elevator is mov+4
  ,n[1 "o c imag9e 2+ a# 6det]m9e :e!r
or n ! elevator 0 mov+ 0he>+ ! s.d (!
motor or 0feel+ xs vibr,n4 ,b if ! motor
is v smoo? or if "o _c he> x1 !n "! re,y
is no way 6tell directly :5 y wake up
:e!r ! elevator is mov+ or n4
,9directly1 "o c m1sure ! value ( s'(t)
0obs]v+ h[ fa/ ! floor numb]s >e *ang+4
,if ! spac+ 2t floors is ab #10 feet1 &
if 9 "o second y drop f floor #17 6floor
                                     #cd
#14, !n r\<ly1 s'(t) .k -#30 ft_/;s4
  ,s f>1 we hav5't yet hit a func;n we c
directly exp]i;e2 b :at ab ! second
derivative1 s''(t)_8 ,! second
derivative is ! rate ( *ange (! veloc;y4
  s''(t) .k "lim%.,dt $o #0] ?s'(t+.,dt)
    -s'(t)_/.,dt#,
s s''(t) w 2 m1sur$ 9 units ( feet p]
second p] second1 or ft_/;s^2_4 ,? rate
( *ange is call$ ! .a3el],n (! elevator4
,a l>ge a3el],n1 say #30 ft_/;s^2, wd
m1n t ea* second1 ! elevator 0 go+ upw>d
at a spe$ ( #30 ft_/;s  fa/] ?an !
previ\s second4 ,an a3el],n l
-#0.03 ft_/;s^2 wd m1n t ea* second !
upw>d veloc;y 0 #0.03 ft_/;s  less ?an x
_h be5 a second 2f4
  ,h[ d y m1sure a3el],n 9 ! elevator8
,well1 y cd e/imate veloc;y at #2 po9ts
& look at xs rate ( *ange1 b ? wd 2
-plicat$ & ]ror-pr"o4 ,b "!'s a simpl]
way4 ,:5 y a3el]ate upw>ds1 y wei< m2 :5
y a3el]ate d[nw>ds1 y wei< less4 ,y c
feel a3el],n 9 ! pit ( yr /oma*1 :e!r y
                                     #ce
c r1d ! floor numb]s or n4
  ,9 fact1 "o c ev5 2 quantitative ab ?
obs]v,n4 ,ac 6,newton1 ,f .k ma, ": ;,f
is ! =ce on an object1 ;m is xs mass1 &
a .k s''(t) is xs a3el],n4 ,s if y "k yr
mass1 & if y plan ah1d & take a ba?room
scale ) y :5 y g 96! elevator s t y c
m1sure quantitatively ! =ce on yr body1
!n y c immly 9f] ! a3el],n4
  ,n[ on 6! ?ird derivative1 s'''(t)_4
,! ?ird derivative is ! rate ( *ange (!
a3el],n4 ,x w "!=e 2 l>ge 9 absolute
value if ! a3el],n *anges rapidly4
,rapid *ange 9 a3el],n wd result1 =
9/.e1 9 y wei<+ a lot "o mo;t1 & v ll !
next2 or 9 y wei<+ v ll "o mo;t &a lot !
next4 ,t is1 ! ?ird derivative w 2 l>ge
dur+ a j]k4 ,j z s'(t) is call$ !
veloc;y & s''(t) is call$ ! a3el],n1
s'''(t) al has a "n---x's call$ ! .j]k1
: is 9t5d$ 6capture xs physical m1n+4 ,s
my fr ,*>lie ,fe6]man 0 "r3 "o c feel !
?ird derivative directly1 &a smoo? ride
m1ns a small ?ird derivative4
                                     #cf
    ,! ,*a9 ,rule
  ,we've se5 9 lab ! la/ situ,n 9 : we
ne$ 6-pute derivatives1 ! case ( func;ns
built 0-pos+ "o func;n ) ano!r 6produce
f(g(x))_4 ,\r 3jecture is t
  ?d_/dx#f(g(x)) .k f'(g(x))g'(x)_4
  ,? 3jecture is 9 fact correct2 x is
call$ 9 ! trade ! ,*a9 ,rule4 ,2f we try
6prove x1 ?\<1 let's pause =a mo;t &
make sure we c p>se ! not,n4 ,on ! left
h& side1 f(g(x)) says 6take f(x) &
6replace ;x by g(x) "ey":4
?d_/dx#f(g(x)) says 6take ! derivative
(! result+ expres.n4 ,s ! left h& side
m1ns 6sub/itute g(x) = ;x & !n
6di6]5tiate4
  ,on ! "r h& side1 f'(g(x)) says 6f/
take ! derivative ( f, & !n 6sub/itute
g(x) = ;x_4 ,! ,*a9 ,rule tells u t !
ord] 9 : we d ! sub/itu;n & di6]5ti,n
matt]s4 ,x is n ! case t
?d_/dx#f(g(x)) .k f'(g(x))_4 ,r1 "o has
6multiply ! "r h& side by g'(x) 9 ord]
6get equal;y4
                                     #cg
  ,! po9t (! ,*a9 ,rule1 ?\<1 is n re,y
t ! ord] 9 : we di6]5tiate & sub/itute
matt]s4 ,r1 x tells u t if we c
di6]5tiate ;f & g, !n we c al di6]5tiate
f(g(x))_4 ,s9ce e func;n "o c write
explicitly seems 6be built up f p[]s (
;x or f s9es & cos9es 0a4i;n1 subtrac;n1
multiplic,n1 divi.n1 or -posi;n (the
process ( =m+ f(g(x)))1 x seems t we h
n[ -plet$ ! process ( le>n+ h[
6di6]5tiate "ey?+4

,a ,mod]n 8,pro(0 (! ,*a9 ,rule4
  ,6prove ! ,*a9 ,rule is actu,y a bit
delicate4 ,let me give :at is ess5ti,y !
mod]n >gu;t1 )a bit ( te*nical detail
left \4 ,if we j plunge boldly 96a
calcul,n1 we wd use ! def9i;n (!
derivative 6write
  ?d_/dx#f(g(x))
    .k "lim%.,dx $o #0] ?f(g(x+.,dx))
    -f(g(x))_/.,dx#_4
  ,x's n 9iti,y cle> :at 6d ) ? limit4
,h[ cd we simplify x at all8 ,an answ]
                                     #ch
turns \ 6be 6try 6bridge ! gap 2t !
num]ator & denom9ator 0multiply+ &
divid+ 0"s?+ algebraic,y 9t]m$iate 2t !
two4 (,we multiply by #1 9 a -plicat$
way_4) ,t is1 we write
  ?d_/dx#f(g(x))
    .k "lim%.,dx $o #0] ?f(g(x+.,dx))
      -f(g(x))_/.,dx#
    .k "lim%.,dx $o #0] ?f(g(x+.,dx))
      -f(g(x))_/g(x+.,dx)-g(x)#*?g(x
      +.,dx)-g(x)_/.,dx#
    .k @("lim%.,dx $o #0] ?f(g(x+.,dx))
      -f(g(x))_/g(x+.,dx)-g(x)#@)*@(
      "lim%.,dx $o #0] ?g(x+.,dx)-g(x)
      _/.,dx#@)
    .k @("lim%.,dx $o #0] ?f(g(x+.,dx))
      -f(g(x))_/g(x+.,dx)-g(x)#@)g'(x)_4
  ,we've n[ made r1l progress4 ,we've
pull$ ! derivative ?d_/dx#f(g(x)) a"p
9to #2 factors1 "o ( : is g'(x), j z !
,*a9 ,rule say x %d 24 ,:at's left 6d is
6>gue t ! rema9+ ugly limit is 9 fact
f'(g(x))_4

                                     #ci
  ,6d ?1 rememb] t if ;g is di6]5tia# at
x, !n ;g is 3t9u\s at ;x_4 ,? m1ns t z
.,dx $o #0, g(x+.,dx) $o g(x)_4 ,6cl1n
up \r not,n a bit1 def9e
  .,dg .k g(x+.,dx)-g(x)
  z .k g(x)_4
,we "k t :5 .,dx $o #0, .,dg $o #0 z
well4 ,s ! rema9+ limit 9 \r expres.n =
?d_/dx#f(g(x)) c 2 rewritt5 z
  "lim%.,dx $o #0] ?f(g(x+.,dx))-f(g(x))
      _/g(x+.,dx)-g(x)#
    .k "lim%.,dx $o #0] ?f(z+.,dg)-f(z)
      _/.,dg#
    .k "lim%.,dg $o #0] ?f(z+.,dg)-f(z)
      _/.,dg#
    .k f'(z) .k f'(g(x))_4
,plu7+ ? 96! =mula abv -pletes ! pro(4
  ,h"o/y -pels me 6po9t \ t "! is a
sli<t pro#m ) ? >gu;t4 ,"! is no r1son
"! %d n 2 po9ts x+.,dx ne> ;x at :
g(x+.,dx)-g(x) .k #0_4 ,at po9ts l ?1 !
expres.n g(x+.,dx)-g(x) we h multipli$ &
divid$ 0is #0, : is a pro#m1 s9ce we c't
divide by #0_4 ,a -pletely rigor\s pro(
                                     #dj
wd ne$ 6a4ress ? 3c]n4 ,-e back 9
,analysis ,a1 & we'll d t4

,a ,classical ,pro( (! ,*a9 ,rule4
  ,! mod]n pro( abv seems 6require "o
6be aw;lly cl"e4 ,is ? re,y :at ! 9itial
>*itects ( calculus did8 ,no1 & x cdn't
h be54 ,newton & ,leibniz didn't ev5 "k
ab limits1 af all4 ,a lot ( 9si<t c 2
ga9$ 0look+ at h[ ,leibniz wd h ?"\ ab !
,*a9 ,rule4 ,f/1 let's rewrite ! ,*a9
,rule 0j *ang+ ! "n ( g, : looks l x cd
only 2 a func;n1 to u, : looks l x cd 2
ei a func;n or a v>ia#4 ,! ,*a9 ,rule wd
!n say 9 "s?+ l ,newton's not,n t
  (f(u(x)))' .k f'(u(x))u'(x)_4
  ,h[ wd ? look 9 ,leibniz,8 not,n8 ,we
"k ,leibniz wd h writt5 u'(x) z
?du_/dx#_4 ,h[ wd he h writt5 ! o!r two
derivatives 9 ! =mula8 ,he wd h be5 r
less explicit ?an mod]n not,n causes "o
6be4 ,he wd h sd1 f .k f(u) .k f(u(x)) c
2 ?\< ( ei z a func;n ( ;u ": we "k ;u
re,y dep5ds on x, or z a func;n ( ;x_4
                                     #da
,reg>d+ ;f z a func;n ( x, ,leibniz wd
write ! derivative (f(u(x)))' z
?df_/dx#_4 ,reg>d+ ;f z a func;n ( u,
,leibniz wd write ! derivative f'(u(x))
z ?df_/du#_4 ,s ! ,*a9 ,rule writt5
0,leibniz wd say
  ?df_/dx# .k ?df_/du#?du_/dx#_4
  ,h[ wd ,leibniz prove ?8 ,he'd c.el !
du & 2 d"o6
  ,9 ,leibniz,8 "u/&+ (! calculus1 ? wd
make p]fect s5se4 ,leibniz reg>d$ !
derivative dy_/dx z re,y 2+ a di6];e
quoti5t .,dy_/.,dx 9 : ! num]ator &
denom9ator 7 bo? 9f9itesim,y small2 s =
hm1 ! derivatives abv 7 re,y frac;ns1 &!
c.ell,n 0 -pletely legitimate4 ,= u1 !
situ,n is m -plicat$4 .,dy_/.,dx is n a
frac;n1 b a limit ( di6];e quoti5ts4 ,x
doesn't re,y h a num]ator &a denom9ator
t c 2 c.ell$4 ,we "!=e ne$ ! m -plex
>gu;t4 ,b if y look ag at \r pro(1 y /
see ! traces ( ,leibniz,8 "u/&+4
,leibniz multiplies & divides by du_4
,9side ! limit1 we multiply & divide by
                                     #db
g(x+.,dx)-g(x) .k .,dg .k .,du (rememb]
t ;g & ;u 7 di6]5t "ns =! same ?+)_4 ,s
,leibniz,8 pro( is / alive 9 ! mod]n
>gu;t4
  ,ev5 ?\< ,leibniz,8 9t]pret,n (
calculus 9 t]ms ( 9f9itesimals isn't !
po/-#1850 "u/&+ (! subject1 >gu;ts tr1t+
dy & dx z if !y 7 numb]s & dy_/dx z if x
7 a frac;n c ne>ly alw 2 rephras$ 9 t]ms
( limits 6yield valid mod]n pro(s4
,9de$1 if ? 7 n s1 !n x wd probably h
be5 nec 6base calculus on "s 3cept o!r
?an t ( limit 9 ord] 6pres]ve ! valid;y
( ^! ,leibniz1n >gu;ts4 ,physici/s &
*emi/s1 economi/s & 5g9e]s1 & o!r
practi;n]s ( calculus o!r ?an pure
ma!maticians (t5 "w & ?9k 9 t]ms ( ^!
9f9itesimals1 "k+ or tru/+ t !
profes.nal ma!maticians cd1 if ne$$1
r5d] rigor\s ! simple >gu;ts !y produce
) ,leibniz0' tools4
  ,leibniz,8 >gu;t =! ,*a9 ,rule al
holds a simple 9tuitive "u/&+ ( :at's
go+ on 9side ! daunt+ look+ expres.n
                                     #dc
f'(u(x))u'(x)_4 ,suppose1 = 3crete;s1 t
;x repres5ts "t1 t ;u repres5ts ! m>ket
value ( "s 9ve/;t1 & t ;f repres5ts !
blood pressure (! [n] ( t 9ve/;t4 ,!
value (! 9ve/;t has 2gun 6drop 9 eag]
trad+1 &! 9ve/or's blood pressure1 :
dep5ds on ! value ( 8 9ve/;t1 has 2gun
6rise4
  ,we "k t ! value ;u (! 9ve/;t is a
func;n (! "t ;x_4 ,we %d "!=e write ;f z
f(u(x)), s9ce ;f dep5ds on ;u : dep5ds
on ;x_4 ,! derivative ?d_/dx#(f(u(x)))
"!=e repres5ts ! rate ( *ange (!
9ve/or's blood pressure z a func;n ( "t4
,! derivative u'(x) repres5ts ! rate (
*ange (! value (! 9ve/;t z a func;n (
"t1 &! derivative f'(u(x)) .k ?df_/du#
repres5ts ! rate ( *ange (! blood
pressure z a func;n (! value4 ,:at !
,*a9 ,rule says is t
  ?df_/dx# .k ?df_/du#*?du_/dx#,
i4e41 t 6f9d h[ fa/ ! 9ve/or's blood
pressure is *ang+ ) "t1 we %d f9d ! rate
( *ange (! value ( 8 9ve/;t z a func;n (
                                     #dd
"t1 & multiply 0! rate ( *ange ( 8 blood
pressure z a func;n (! value ( 8 9ve/;t4
,? seems 6me 6be em95tly r1sona#4

,examples (! ,*a9 ,rule4
  ?d_/dx#(sin (x^2")) .k cos (x^2")*2x_4
(,"h f(x) .k sin x & g(x) .k x^2_4)
  ?d_/dx#(sin^2 x) .k #2sin xcos x_4
(,"h f(x) .k x^2 & g(x) .k sin x_4)
  ?d_/dx#((?3x+5_/x^2"+1#)^100")
    .k #100(?3x+5_/x^2"+1#)^99"*?(x^2"
    +1)3-(3x+5)2x_/(x^2"+1)^2"#_4
(,"h f(x) .k x^100 &
g(x) .k ?3x+5_/x^2"+1#_4)
  ?d_/dx#(x^2"sin (>x])
    .k #2xsin (>x])+x^2"cos (>x])
    *?1_/2>x]#_4
(,"h we ne$$ 6use f/ ! ,product ,rule
&!! ,*a9 ,rule_4)
  ?d_/dx#(f(g(h(x))))
    .k f'(g(h(x)))*?d_/dx#(g(h(x)))
    .k f'(g(h(x)))g'(h(x))h'(x)_4
,= 9/.e1
  ?d_/dx#(sin >x^3"+11])
                                     #de
    .k cos >x^3"+11]*?1_/2>x^3"+11]#
    *3x^2_4
  ?d_/dx#(>x+.>x+..>x..].]])
    .k ?1_/2>x+.>x+..>x..].]]#(1+?1_/2>x
    +.>x.]]#(1+?1_/2>x]#))_4
  ,if we c -pute ! derivative ( ?1 !n we
re,y c -pute ! derivative ( j ab any?+4

    ,implicit ,di6]5ti,n & ,relat$
    ,rates
  ,^! >e two 9t]e/+ id1s t repres5t side
li<ts 9 \r curr5t que/ = "u/&+1 b t >e
wor? ca/+ a gl.e at z we g pa/ !m4 ,"!
>e situ,ns 9 : ea* is critic,y
important4

,implicit ,di6]5ti,n
  .,a .f/ .example

  ,! simple/ way 6write ! equ,n (! unit
circle 9 ,c>tesian coord9ates is
x^2"+y^2 .k #1_4 ,suppose we want$ 6"k !
slope (! tang5t l9e 6? curve at ! po9t
(?3_/5#, ?4_/5#)_4 (,? is actu,y easy
                                     #df
6get f ele;t>y geometry1 b be> ) me =!
example_4)
  ,! obvi\s naive approa* wd 2 6solve =
;y &! di6]5tiate4 ,we'd get
  y .k >1-x^2"],
s t
  y' .k ?1_/2>1-x^2"]#(-2x)
    .k -?x_/>1-x^2"]#_4
,at x .k ?3_/5#,
y' .k -,?3_/5,_/>1-(3_/5)^2"],# .k -?3_/4#
_4
  ,"! >e two pro#ms ) ? naive approa*4
,f/1 ! 9itial equ,n wdn't h 6get m* m
-plicat$ ?an x^2"+y^2 .k #1 2f x 2came
di6icult or impossi# 6solve4 ,second1 !
fact t we get a -plicat$ expres.n :5 we
solve = ;y m1ns t \r expres.n = y' is
gu>ante$ 6be a mess4 ,is "! any way
6resolve ^! 3c]ns8
  ,yes4 ,"!'s a cl"e second way 6-pute !
slope 9 situ,ns l ?1 call$ ,implicit
,di6]5ti,n4 ,! idea is 6/>t )! orig9al
equ,n x^2"+y^2 .k #1 & 6di6]5tiate "ey?+
9 si<t )\t bo!r+ 6solve = ;y f/4 ,:5 we
                                     #dg
d di6]5tiate1 ?\<1 we h 6rememb] t ;x is
j ! v>ia#1 b t y .k y(x) is re,y a
func;n ( ;x_4 ,? m1ns t ! derivative (
;x is #1, b t ! derivative ( ;y is y'_4
,:5 we di6]5tiate1 we "!=e get
  x^2"+y^2 .k #1
  (x^2"+y^2")' .k #1'
  (x^2")'+(y^2")' .k #1'
  #2x+2yy' .k #0_4
  ,if t looks 3fus+1 !n y cd write
explicitly t y .k f(x), s t ! la/ #2 l9e
(! calcul,n wd look l
  (x^2")'+(f(x)^2")' .k #1'
  #2x+2f(x)f'(x) .k #0,
: is a /rai<t=w>d applic,n (! ,*a9
,rule4
  ,n[ t we've d"o ! di6]5tiat+ 6get
#2x+2yy' .k #0, x's easy 6solve 6get
  y' .k -?x_/y#_4
,we want$ ! derivative at ! po9t
(?3_/5#, ?4_/5#), : m/ "!=e 2
  y' .k -?x_/y# .k -,?3_/5,_/4_/5,#
    .k -?3_/4#,
! same result we got 0f/ solv+ = ;y & !n
                                     #dh
di6]5tiat+4

  .,a .second .example

  ,6see j h[ p[];l ! te*nique (
,implicit ,di6]5ti,n is1 let's try a
second example4 ,suppose we _h a curve
giv5 0! equ,n
  x^2"y^5"-3xy^2"+x-y+2 .k #0_4
,we >e ask$ 6f9d ! slope (! tang5t l9e
6? curve at ! po9t (1, -1)_4
  ,n[1 ! />t+ equ,n is a #5th degree
equ,n = ;y_4 ,if x _h be5 a quadratic =
y, we cd h us$ ! ,quadratic ,=mula
6solve x4 ,if x _h be5 ev5 an equ,n (
degree #3 or #4, we cd h us$ =mulas l !
,quadratic ,=mula due 6! #16th ,c5tury
,italian ma!maticians ,t>taglia &
,c>dano 6solve x 9 t]ms ( ?ird & f\r?
roots2 ?\< ! solu;ns wd take s"eal pages
6write4 ,we'd !n h 6di6]5tiate !m4 ,b
fif? degree equ,ns >e ev5 worse4 ,2f 8
d1? at #27, ! ,norwegian ma!matician
,abel prov$ t a g5]al #5th degree equ,n
                                     #di
_c 2 solv$ 9 radicals---"! is'nt a =mula
l ! ,quadratic ,=mula = solv+ equ,ns (
degree #5 or hi<]4 ,ev5 /r;g] & m* deep]
results ab solvabil;y 7 sket*$ 0,ev>i/e
,galois 2f 8 d1? 9 a duel at age #20, &
h gr[n 96a ri* & b1uti;l 4cipl9e call$
,galois ,!ory2 b ,i'm n[ gett+ 4tract$4
,! critical po9t = u "r n[ is t ,i c't
solve ! equ,n abv = y, & nei c y4 ,\r
9itial approa* 6di6]5tiat+ x "!=e c 2
prov5 n 6"w4
  ,) ,implicit ,di6]5ti,n1 ?\<1 x's /
easy4 ,f/1 take ! derivative (! equ,n1
rememb]+ t ;x is ! v>ia# & t ;y is a
func;n ( ;x_4 ,y get
  #2xy^5"+x^2"(5y^4"y')-3y^2"-3x(2yy')+1
    -y' .k #0_4
,n[ gr\p ! t]ms 3ta9+ y', & solve4
  (5x^2"y^4"-6xy-1)y' .k -#2xy^5"+3y^2"-
    1
  y' .k ?-2xy^5"+3y^2"-1_/5x^2"y^4"-6xy-
    1#_4
,we want 6"k ! value ( y' at ! po9t
(1, -1), s we replace ;x ) #1 & ;y ) -#1
                                     #ej
& we get
  y' .k ?2+3-1_/5+6-1# .k ?2_/5#_4
,is ? amaz+1 or :at8

  ..,did ,i j .*1t8

  ,implicit ,di6]5ti,n is an amaz+
te*nique1 b ,i h p]h made x seem ev5 m
amaz+ ?an x is4 ,i sd t ! equ,n
x^2"y^5"-3xy^2"+x-y+2 .k #0 cd n 2 solv$
= ;y_4 ,if ? is s1 !n h[ 0 x possi# 6f9d
! ;y coord9ate (! po9t (1, -1)_8 ,didn't
,i j *1t8
  ,n re,y4 ,9 say+ t we c't solve
x^2"y^5"-3xy^2"+x-y+2 .k #0, :at ,i m1n
6say is t we c't f9d a =mula = ;y z a
func;n ( x, : is :at we ne$ 9 ord] 6be
a# 6di6]5tiate ;y_4 ,b if we "k ! value
( x, say1 x .k #1, !n ! equ,n simplifies
to
  y^5"-3y^2"-y+3 .k #0_4
,6f9d num]ical values ( ;y satisfy+ ?
equ,n1 we cd d "st+ z simple z graph+ !
func;n & zoom+ 9 on ! po9ts ": x crosses
                                     #ea
! axis4 ,? is way simpl] ?an solv+ !
orig9al equ,n analytic,y4 ,9 fact1 a
graph ( f(y) .k y^5"-3y^2"-y+3 (with !
;y-axis horizontal) is %[n 9 ,figure
#16_4 ,x's easy 6see f ! ,figure t "! is
a root v close to y .k -#1, z well z two
positive roots4

,relat$ ,rates
  ,implicit ,di6]5ti,n is a te*nique =
f9d+ ! derivative y' 9 ! situ,n ": !
v>ia# ;x &! func;n ;y >e relat$ 0"s
equ,n4 ,! idea ( ,relat$ ,rates is t if
9/1d (a func;n &a v>ia#1 "! >e two
func;ns relat$ 0"s equ,n1 !n !
derivatives ( ^! func;ns >e al relat$4

  .,"s ."qs

  ,3sid] ! foll[+ situ,n4 ,a p]son #5
feet tall is 9 a li<t$ p>k+ lot at ni<t4
,! lot 3ta9s a s+le /reet li<t on a #15
foot tall pole4 ,z ! p]son walks away f
! li<t1 h] %ad[ gets l;g] & l;g]4 ,!
                                     #eb
p]son is n necess>ily walk+ at a 3/ant
spe$1 b :5 %e is #30 feet f ! li<t1 %e
is mov+ away f ! li<t at #4 ft_/;s4 ,h[
fa/ is ! l5g? (! p]son's %ad[ gr[+ at ?
mo;t1 : is sket*$ 9 ,figure #17_8
  ,"o c ask ano!r1 relat$ "q t ,i alw
f9d 9trigu+4 ,suppose ! p]son's veloc;y
does happ5 6be 3/ant at #4 ft_/;s  away
f ! li<t4 ,does ! l5g? (! %ad[ gr[ qkly
:5 ! p]son is close 6! li<t & !n m sl[ly
lat]1 or does x gr[ fa/] & fa/] z !
p]son moves away f ! li<t1 or does x gr[
at a 3/ant rate8 ,al? ,i've walk$ away f
an aw;l lot ( /reet li<ts 9 my life1 ,i
f9d x r h>d 6answ] ? "q 0?9k+ back on my
exp]i;e4 ,9de$1 ,i c imag9e a v>iety (
3v9c+ s.d+ answ]s3
  ,:5 ! p]son is close 6! /reet li<t1 !
angle f ! p]son'r h1d 6! lamp 6! pole is
9cr1s+ rapidly1 : m1ns ! %ad[ is 9cr1s+
rapidly4 ,:5 ! p]son is f> f ! li<t1 ?
angle is h>dly *ang+ at all1 s ! l5g? (!
%ad[ probably isn't *ang+ m*4 ,s ! l5g?
(! %ad[ gr[s fa/ at f/1 !n sl[ly4
                                     #ec
  ,or try ?3 ,:5 ! p]son is f> f ! li<t1
y "k ! %ad[ is re,y l;g4 ,x m/ 2 gr[+
pretty fa/ 6h gott5 t l;g4 ,fur!r1 ev5 a
t9y *ange 9 ! angle w make a huge di6];e
9 ! l5g? (a re,y l;g %ad[4 ,s ! l5g? (!
%ad[ gr[s sl[ly at f/1 !n fa/4
  ,or maybe "sh[ "ey?+ c.els1 &! rate (
gr[? is a 3/ant4
  ,: analysis1 if any1 is "r8

  .,"s .answ]s

  ,let's />t )! f/ "q1 ab ! rate ( gr[?
(! %ad[ at ! mo;t ! p]son is #30 feet f
! li<t4 ,"! >e re,y two func;ns 9volv$
"h3 p(t), ! 4t.e f ! li<t po/ 6! p]son
at "t t, & s(t), ! l5g? (! %ad[ at "t
;t_4 ,^! func;ns >e relat$ 0! fact t !
trangles 9 ! picture >e simil>4 ,! small
triangle1 ^: v]tical side is ! #5 foot
p]son1 has horizontal side s(t)_4 ,!
l>ge triangle1 ^: v]tical side is ! #15
foot li<t po/1 has horizontal side
p(t)+s(t)_4 ,we "!=e h
                                     #ed
  ?s(t)_/5# .k ?p(t)+s(t)_/15#_4
  ,! pro#m tells u p(t) & p'(t) at a
"picul> "t t, & asks u 6f9d s'(t) at t
"t4 ,6d ?1 we di6]5tiate ! equ,n abv
6get
  ?s'(t)_/5# .k ?p'(t)+s'(t)_/15#_4
,s9ce we "k t at ! mo;t 9 "q1
p'(t) .k #4 ft_/;s , we h
  ?s'(t)_/5# .k ?4+s'(t)_/15#,
: solves 6give s'(t) .k #2_4 ,! %ad[ is
gr[+ at #2 ft_/;s4
  ,9 fact1 ? calcul,n al answ]s ! second
"q abv1 ab h[ ! rate ( gr[? (! %ad[
*anges ) "t4 ,! expres.n we j got =
s'(t), "nly s'(t) .k #2, doesn't 3ta9
p(t) at all4 ,z l;g z ! p]son keeps
walk+ at #4 ft_/;s , ! analysis abv is
valid1 & s'(t) .k #2 ft_/;s4 ,! rate (
gr[? (! %ad[ is "!=e a 3/ant1 reg>d.s (
h[ f> ! p]son is f ! li<t4

  .,ano!r .example


                                     #ee
  ,relat$ ,rates seem 6be a popul> topic
) au?ors ( calculus texts1 9 "p 2c !y
give rise 6lots ( 5t]ta9+ ^w pro#ms4
,let me j d "o m "h4
  ,"p (a class project n assign$ 0!
professor1 & n yet 9volv+ ! professor1
requires fill+ a re,y big wat] balloon4
,? is a ^w pro#m1 n r1l life---& ,i'd l
y 6keep x t way---s ! balloon is assum$
6be a p]fect sph]e at all "ts4 ,! tap f
: ! balloon is 2+ fill$ produces #1
cubic foot ( wat] p] m9ute4 ,h[ fa/ is !
radius (! balloon 9cr1s+ :5 ! radius is
#3 9*es8 #6 9*es8
  ,6solve ? pro#m1 y ne$ 6rememb] t !
radius & volume (a sph]e >e relat$ 0!
equ,n
  ,v .k ?4_/3#.pr^3_4
,! pro#m seems 6be 3sid]+ bo? ! radius
&! volume z func;ns ( "t1 s maybe we %d
re,y write
  ,v(t) .k ?4_/3#.pr(t)^3_4
  ,if we di6]5tiate "ey?+ 9 si<t1 we get
  ,v' .k ?4_/3#.p(3r^2"r')
                                     #ef
    .k #4.pr^2"r'_4
,we >e told t ,v' .k #1 ft^3"_/ min
.k #1728 in^3"_/ min4 ,6f9d ! rate (
*ange (! radius :5 r .k #3 in , we j
plug 9 ^? values & solve3
  #1728 .k #4.p(9r')
  r' .k ?1728_/36.p# .k ?48_/.p#
    ".k<*] #15.279 in_/min4
,:5 r .k #6 in , we h
  #1728 .k #4.p(36r')
  r' .k ?1728_/144.p# .k ?12_/.p#
    ".k<*] #3.8197 in_/min4
  ,x's a gd ex]cise 9 ! develop;t ( yr
9tui;n 6see if y c 3v9ce yrf 9tuitively
t x makes s5se t ! rate ( *ange (!
radius %d 2 less :5 r .k #6 ?an :5
r .k #3_4 ,:y %d x 2 exactly "o qu>t] z
grt8






                                     #eg