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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "Calculus I" }}{PARA 260 "
" 0 "" {TEXT -1 33 "Lesson 7:  Differentiation Rules " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 186 "In this works
heet, we will review the definition of the derivative of a function, l
ook at Maple's commands for differentiation, and use them to verify th
e basic differentiation formulas." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 32 "The definition of the Derivative" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 52 "First, the definition.  The derivativeof a function " }
{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 14 " at the point " }{XPPEDIT 
18 0 "x;" "6#%\"xG" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {XPPEDIT 18 
0 "D(f)(x) := limit((f(x+h)-f(x))/h,h = 0);" "6#>--%\"DG6#%\"fG6#%\"xG
-%&limitG6$*&,&-F(6#,&F*\"\"\"%\"hGF3F3-F(6#F*!\"\"F3F4F7/F4\"\"!" }
{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 87 "provided, of course, th
at the limit exists.  Since the derivative depends on the point " }
{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 50 " where it is evaluated, it \+
is itself a function.  " }{TEXT 256 5 "Maple" }{TEXT -1 19 " uses the \+
notation " }{XPPEDIT 18 0 "D(f);" "6#-%\"DG6#%\"fG" }{TEXT -1 47 " for
 this function; the more usual notation is " }{XPPEDIT 18 0 "f;" "6#%
\"fG" }{TEXT -1 69 " ' .  You should by now be familiar with several i
nterpretations for " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 47 " ' : \+
it is the instantaneous rate of change of " }{XPPEDIT 18 0 "f;" "6#%\"
fG" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 54 "; i
t is the slope of the tangent line to the graph of " }{XPPEDIT 18 0 "f
;" "6#%\"fG" }{TEXT -1 5 "; if " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"
xG" }{TEXT -1 46 " represents the position of an object at time " }
{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "f;
" "6#%\"fG" }{TEXT -1 3 " '(" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 
18 ") is its velocity." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }
{TEXT 273 7 "Maple's" }{TEXT -1 25 " differentiation commands" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 22 "It is easy to write a " }{TEXT 256 5 "Maple" }
{TEXT -1 147 " procedure which computes the derivative of a function u
sing the definition.  (The command unapply is used to convert an expre
ssion to a function.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "my
D := proc(f) unapply(limit((f(x+h)-f(x))/h,h=0), x) end:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f := x -> x^2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"#\"\"
\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "myD(f);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF&,$9$
\"\"#F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "myD(sin);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%$cosG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 30 "However, the built-in command " }{TEXT 257 1 "D" }{TEXT 
-1 21 " does the same thing:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 5 "D(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG
%&arrowGF&,$9$\"\"#F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 
"D(sin);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%$cosG" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 10 "Note that " }{TEXT 258 1 "D" }{TEXT -1 19 " diffe
rentiates a  " }{TEXT 259 9 "function," }{TEXT -1 74 " and that the de
rivative of a function is again a function.  For example, " }{XPPEDIT 
18 0 "D(f);" "6#-%\"DG6#%\"fG" }{TEXT -1 29 " can be evaluated at a po
int:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "D(f)(3); D(f)(-5); \+
D(f)(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "but " }{TEXT 262 5 "Maple" }{TEXT 
-1 83 " provides another command which is closer to ordinary mathemati
cal practice (it is " }{TEXT 263 7 "Maple's" }{TEXT -1 19 " equivalent
 of the " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT 
-1 25 " notation.)  The command " }{TEXT 264 4 "diff" }{TEXT -1 19 " d
ifferentiates an " }{TEXT 274 11 "expression," }{TEXT -1 35 " and give
s back another expression." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "diff(x^2, x); diff(x^3 + 5*x, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,$%\"xG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"#\"\"
\"\"\"$\"\"&F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Notice that whe
n you use " }{TEXT 265 0 "" }{TEXT -1 0 "" }{TEXT 266 4 "diff" }{TEXT 
-1 16 ", you must tell " }{TEXT 267 5 "Maple" }{TEXT -1 101 " the inde
pendent variable.  Guess what the answer to the next command will be b
efore you hit <enter>." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "d
iff(x^2,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 42 "That's all there is to differentiation in " }
{TEXT 268 5 "Maple" }{TEXT -1 163 ".  Just remember that the derivativ
e of a function is another function, and the derivative of an expressi
on is another expression, and that they are computed with " }{TEXT 
269 1 "D" }{TEXT -1 5 " and " }{TEXT 270 4 "diff" }{TEXT -1 42 " respe
ctively.  This is a nice example of " }{TEXT 271 5 "Maple" }{TEXT -1 
134 " forcing you to think clearly, by the way.  The distinction betwe
en differentiating functions and expressions was not invented by the \+
" }{TEXT 272 5 "Maple" }{TEXT -1 110 " programmers: it is a real one, \+
and is reflected in the fact that we have both the prime (') notation \+
and the " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT 
-1 86 " notation for differentiation, but it is usually glossed over o
r ignored in textbooks." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 21 "Diff
erentiation rules" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Now let's use
 " }{TEXT 256 1 "D" }{TEXT -1 249 " and diff to compute some derivativ
es.  While doing this, we will take the opportunity to verify three of
 the basic differentiation rules.  (The remaining one is the Chain Rul
e, which we will explore in another worksheet.)  First, the derivative
 is " }{TEXT 276 8 "linear: " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "diff(
a*f(x)+b*g(x),x) = a*diff(f(x),x)+b*diff(g(x),x);" "6#/-%%diffG6$,&*&%
\"aG\"\"\"-%\"fG6#%\"xGF*F**&%\"bGF*-%\"gG6#F.F*F*F.,&*&F)F*-F%6$-F,6#
F.F.F*F**&F0F*-F%6$-F26#F.F.F*F*" }{TEXT -1 20 "  for any constants " 
}{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b;" 
"6#%\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "Second, we ha
ve the " }{TEXT 275 13 "product rule:" }}{PARA 258 "" 0 "" {XPPEDIT 
18 0 "diff(f(x)*g(x),x) = diff(f(x),x)*g(x)+f(x)*diff(g(x),x);" "6#/-%
%diffG6$*&-%\"fG6#%\"xG\"\"\"-%\"gG6#F+F,F+,&*&-F%6$-F)6#F+F+F,-F.6#F+
F,F,*&-F)6#F+F,-F%6$-F.6#F+F+F,F," }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" 
{TEXT -1 11 "Third, the " }{TEXT 277 14 "quotient rule:" }}{PARA 259 "
" 0 "" {XPPEDIT 18 0 "diff(f(x)/g(x),x) = (diff(f(x),x)*g(x)-f(x)*diff
(g(x),x))/(g(x)^2);" "6#/-%%diffG6$*&-%\"fG6#%\"xG\"\"\"-%\"gG6#F+!\"
\"F+*&,&*&-F%6$-F)6#F+F+F,-F.6#F+F,F,*&-F)6#F+F,-F%6$-F.6#F+F+F,F0F,*$
-F.6#F+\"\"#F0" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
35 "The linearity rule is the simplest." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "diff(x^2, x); diff(x^3, x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$%\"xG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%
\"xG\"\"#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "dif
f(7*x^2 + 4*x^3, x); diff(5*x^2 - Pi*x^3, x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&%\"xG\"#9*&\"#7\"\"\")F$\"\"#F(F(" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&%\"xG\"#5*(\"\"$\"\"\"%#PiGF()F$\"\"#F(!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Here it is again, using " }{TEXT 
278 1 "D" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 
"f := x -> cos(x^2); g := x -> x*exp(2*x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%$cosG6#*$)9$
\"\"#\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"
6$%)operatorG%&arrowGF(*&9$\"\"\"-%$expG6#,$F-\"\"#F.F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "D(f); D(g);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF&,$*&-%$sinG6#*$)
9$\"\"#\"\"\"F3F1F3!\"#F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%
\"xG6\"6$%)operatorG%&arrowGF&,&-%$expG6#,$9$\"\"#\"\"\"*(F0F1F/F1F+F1
F1F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "D(3*f - g);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&R6#%\"xG6\"6$%)operatorG%&arrowGF',$
*&-%$sinG6#*$)9$\"\"#\"\"\"F4F2F4!\"#F'F'F'\"\"$R6#F&F'F(F',&-%$expG6#
,$F2F3F4*(F3F4F2F4F:F4F4F'F'F'!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 25 "If you look carefully at " }{TEXT 279 7 "Maple's" }{TEXT -1 
133 " syntax, you can see that this last answer verifies the rule, but
 it might be easier to check by evaluating the various functions at " 
}{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "D(3*f - g)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,(*&-%$sinG6#*$)%\"xG\"\"#\"\"\"F,F*F,!\"'-%$expG6#,$F*F+!\"\"*(F+F,
F*F,F.F,F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "3*D(f)(x) - D
(g)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&-%$sinG6#*$)%\"xG\"\"#
\"\"\"F,F*F,!\"'-%$expG6#,$F*F+!\"\"*(F+F,F*F,F.F,F2" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "N
ow let's look at the product and quotient rules, with the same functio
ns " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 
"g;" "6#%\"gG" }{TEXT -1 26 ".  The product rule first:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "D(f*g);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&R6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&-%$sinG6#*$)9
$\"\"#\"\"\"F5F3F5!\"#F(F(F(F5%\"gGF5F5*&%\"fGF5R6#F'F(F)F(,&-%$expG6#
,$F3F4F5*(F4F5F3F5F=F5F5F(F(F(F5F5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "D(f*g)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(-%$
sinG6#*$)%\"xG\"\"#\"\"\"F,F)F,-%$expG6#,$F*F+F,!\"#*&-%$cosGF'F,,&F-F
,*(F+F,F*F,F-F,F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "D(
f)*g + f*D(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&R6#%\"xG6\"6$%)o
peratorG%&arrowGF(,$*&-%$sinG6#*$)9$\"\"#\"\"\"F5F3F5!\"#F(F(F(F5%\"gG
F5F5*&%\"fGF5R6#F'F(F)F(,&-%$expG6#,$F3F4F5*(F4F5F3F5F=F5F5F(F(F(F5F5
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "D(f)(x)*g(x) + f(x)*D(g
)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(-%$sinG6#*$)%\"xG\"\"#\"
\"\"F,F)F,-%$expG6#,$F*F+F,!\"#*&-%$cosGF'F,,&F-F,*(F+F,F*F,F-F,F,F,F,
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "In terms of expressions, the \+
same calculations look like" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "diff(f(x)*g(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(-%$sinG
6#*$)%\"xG\"\"#\"\"\"F,F)F,-%$expG6#,$F*F+F,!\"#*&-%$cosGF'F,F-F,F,**F
+F,F3F,F*F,F-F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "diff(f
(x), x)*g(x) + f(x)*diff(g(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
,&*(-%$sinG6#*$)%\"xG\"\"#\"\"\"F,F)F,-%$expG6#,$F*F+F,!\"#*&-%$cosGF'
F,,&F-F,*(F+F,F*F,F-F,F,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "I
n either form , you can see that the product rule works.  Now for the \+
quotient rule:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diff(f(x)
/g(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&-%$sinG6#*$)%\"xG\"
\"#\"\"\"F,-%$expG6#,$F*F+!\"\"!\"#*&-%$cosGF'F,*&F)F,F-F,F1F1*&*&F+F,
F4F,F,*&F*F,F-F,F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sim
plify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&,(*&-%$sinG6#*$)%\"
xG\"\"#\"\"\"F/F,F/F.-%$cosGF*F/*(F.F/F0F/F-F/F/F/-%$expG6#,$F-!\"#F/F
/*$F,F/!\"\"F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "(diff(f(x
), x)*g(x) - f(x)*diff(g(x), x)) / (g(x))^2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&,&*(-%$sinG6#*$)%\"xG\"\"#\"\"\"F-F*F--%$expG6#,$F+F,
F-!\"#*&-%$cosGF(F-,&F.F-*(F,F-F+F-F.F-F-F-!\"\"F-*&F*F-)F.F,F-F8" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,$*&*&,(*&-%$sinG6#*$)%\"xG\"\"#\"\"\"F/F,F/F.-%$c
osGF*F/*(F.F/F0F/F-F/F/F/-%$expG6#,$F-!\"#F/F/*$F,F/!\"\"F9" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "(You can do the calculation with \+
" }{TEXT 280 1 "D" }{TEXT -1 14 " if you wish.)" }}}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 2 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }