,calc ,a1 ,lab #5

    ,curr5t ,/atus
  ,f class & f ! la/ lab1 we "k t
  (f(x)+-g(x))' .k f'(x)+-g'(x)
  (f(x)g(x))' .k f(x)g'(x)+f'(x)g(x)
  (?f(x)_/g(x)#)' .k ?g(x)f'(x)
    -f(x)g'(x)_/g(x)^2"#_4
,^! rules1 tgr )! rules t
  ?d_/dx#(x^n") .k nx^n-1
  ?d_/dx#sin x .k cos x
  ?d_/dx#cos x .k -sin x
let u di6]5tiate a l>ge numb] ( func;ns3
any?+ built up f p[]s ( ;x or f sin x &
cos x 0any -b9,n ( a4i;n1 subtrac;n1
multiplic,n1 & divi.n4
  ,"! is "o situ,n ^! rules d n yet
cov]1 ?\<4 ,:at if we build new func;ns
0apply+ old func;ns "o af ano!r8 ,t is1
suppos+ we "k ! derivatives ( f(x) &
g(x), h[ wd we f9d ! derivative (
f(g(x))_8


                                      #j
    ,examples4
  ,examples ( func;ns built up ? way
mi<t 2
  (a) ,if f(x) .k >x] & g(x) .k x^2"+2,
!n f(g(x)) .k >x^2"+2]_4
  (;b) ,if f(x) .k x^2"-3x+4 &
g(x) .k #2x-5, !n
  f(g(x)) .k (2x-5)^2"-3(2x-5)+4_4
  (;c) ,if f(x) .k x^100 &
g(x) .k x^2"-4x+5, !n
f(g(x)) .k (x^2"-4x+5)^100_4
  (;d) ,if f(x) .k sin x & g(x) .k x^3,
!n f(g(x)) .k sin (x^3")_4
  (;e) ,if f(x) .k x^3 & g(x) .k sin x,
!n f(g(x)) .k (sin x)^3 .k sin^3 x_4
  (;f) ,if f(x) .k >x] &
g(x) .k #1-cos^2 x, !n
f(g(x)) .k >1-cos^2 x]_4
  (;g) ,on a fanci] level1 a func;n l
  x+>x+.>x+..>x..].]]
is al built up 0-posi;n z well z 0a4i;n4
  ,"s ( ^! func;ns1 l (;b)1 (;c) & (;e)
c 2 di6]5tiat$ 0f/ exp&+ !m \ & !n us+ !
rules we h2 o!rs we cd h&le only 0go+
                                      #a
back 6! def9i;n 9volv+ limits4 ,wdn't x
2 nice if we _h a g5]al =mula =
?d_/dx#(f(g(x)))_8

    ,pro#ms4
  #1_4 ,a plausi# 3jecture wd 2 t if
k(x) .k f(g(x)), !n k'(x) .k f'(g(x))_4
,try ? rule \ 0-put+ k'(x) & f'(g(x)) =!
foll[+ pairs ( func;ns3
  (a) f(x) .k x^2, g(x) .k #7x+2_4
  (;b) f(x) .k x^2, g(x) .k x^2_4
  (;c) f(x) .k x^2, g(x) .k x^3_4
  (;d) f(x) .k x^2, g(x) .k x^4_4
  (;e) f(x) .k x^2, g(x) .k x^2"-4x+5_4
  (;f) f(x) .k x^2"-3x+4, g(x) .k #2x-5
_4
  (;g) f(x) .k x, g(x) .k  a g5]ic g(x)
_4
  #2_4 ,bas$ on :at y le>n$ 9 ,pro#m #1,
make a 3jecture ab h[ 6-pute !
derivative ( k(x) .k f(g(x))_4 ,te/ ?
3jecture 0try+ x \ ) f(x) .k >x] &
g(x) .k x^2"+2_4 ,t is1 "w \ :at yr
=mula predicts ! derivative 6be1 & !n
                                      #b
see :e!r yr =mula su3ess;lly calculates
! slopes ( tang5t l9es1 or ! num]ical
limits ( di6];e quoti5ts4
  #3_4 ,once y h a "w+ =mula 9 ,pro#m #2
, use x 6-pute ! derivatives (! func;ns
(a)--(;g) 9 ! ,examples sec;n4 ,if y're
n sure ( yr results1 try look+ at tang5t
l9es graphic,y or num]ic,y 6te/ yr
analytic results4
  ,if y f9i% e>ly1 y cd alw "w on
,mon"d's home"w4













                                      #c