{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 105 " \+ \+ " }{TEXT 256 10 "Calculus I" }{TEXT -1 105 "\n \+ \+ " }}{PARA 0 "" 0 "" {TEXT -1 327 " \+ \+ Derivative Notation and Differentiation Rules \n\nIn 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." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 257 32 "The definition of the Derivative" }} {PARA 0 "" 0 "" {TEXT -1 72 "First, the definition. The derivativeof \+ a function f at the point x is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 81 " \+ " }{OLE 1 3585 1 "[xm]Br=WfoRr B:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::wyyyqy;::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::NDYmq^H;C:ELq^H_mvJ::::::::gjvb?f_nqCvr;V :>j=B:<:=ja^GE=;:::::::::N;?R:yyyyyy:>:<::::::JDJ:j:VBYmp>HYLkNG>::::: :::N:;::::::::_Z:vyyuy:>:<:::: ::AJ:n:>:nYN::wAJ=j=B:KJ:F;N;;j?J@j@>:wAyA:::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::F:DJ:::::::::::s:?jyAvHJ:JfXpD^eSa<>:AR:=r:Ob<=bFA fSTjr:=a:NZ:^=>:EZ :F[=f:V[Z<>f=N\\:B :;xyyQ]yyyyYJHZ:nr;j?<:G;Sj`@Pt\\Pd`QrP@[LHB:qi:;fyB:>l;fB]E:emj^HMmqn G;KaFFJufF^:NZ:vyyuy:>:;B::::::^:FA=?R:AJ:^:vYxY:B::::::f::::::::::J?B:yay=J:B:::::: nYyA<::::::::::::jysy:>:<::::::::JcTTUUSaEBWTSiEB_tUUURWmXmTtYQ]mTt Yq:>Z:>:[K<<:US:>;;JSdJj@JLEj:>Z:F_s?;N@Nt;F:>is?;N@Nb;F:>_s?;N@^`:F:; Jm`QHjw;<:[N:b:DJ::s:qQBv:>:sg::[ K:C:[q:V::B>N:F:nyy yyy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::fyyyyya:nYf::G:jy;::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n:;`:Z@[::Jj]PjPLYaj]Aj;JZPZ:B:F:YLpfF>:::::::::J?NZ;vyy yyy=J:B:::::::c:;:=j[vGUMrvC?MoJ::::::::JCNZ;F:f:vYxY:B::::::F:;JmJ:j; vCj^nGGmq>:;::::::::_Z:vyyuy:>:<::::::AJ:n:>:nYN::wyyyqj>J?>:Q: S:UJ:n;;jA>:[Z:Fgny;;J:::::::F:wyyAbR<:TnEB:UTTAeVYuVYeScEBETVeURcUTYeU;sFWCFl:LncZ;Jn>:;b:gFDDJEa\\:>Z;:?b[_VB?B\\[[:::::::::jysyA:C:=B:;B:yayQZ:J :JCy:MX;Z:F:MZ=>j;B:AB:s:;jDj\\Fh:fH=MtFGYMq>>Wlj:gml>:;:::::JJ Z::::::J`:J:<:::::::>=?R:AJ:^:vYxY:B: :::::v:;J[H:<:=ja:;B:;:::::::::N;:<::::::wqy[:::::::::::::vYxI: ;Z:::::::::f:NZ;F:=f:V[ryyYowyyyy;Y=Ny>j?^;UTRcETcTX[US>^;>;;JwEJ:><<:^:f:;jCDJIDJQpJHB:qi:;fy>Z:JBA:;B:Cb:;B>cTTUUSaEBWTSiEB_tUUURWm XmTtYQ]mTtYq:>:;JR^Z:jPF:K:_cJSJ_ij:Jv`Q>JSJWhj:JN`Q>JSJ:ej:Jm`Q>JSJS]j:jN:_;;y;=:K=N@> q?F:>IJSJNPj:JN:_;WE;=:Gm>B:_KyDJTV:;B:SUTUUSsJ:VYZ:JB[<>ZaTXDpql`q ?B:[KJS>u=>e GF:>yc?;N@KW:kT==:gYTK:_KI_l=F:vvc?;N@kPDIj:JnaP>JS>Ocv:=:GYTK:_K@OhCF :f_c?;N@nr@F:;jO`P?JMJ?vYxy:J:^=VY[j=>:;JXE:;B:=J;Dlc`qsLqlp@;B:OJ:^:> :=B::;JR^:f_;F:C:[Q;<::::::::::::::::::::::::::::::::: ::::::::::2:" }}{PARA 0 "" 0 "" {TEXT -1 288 "provided, of course, tha t the limit exists. Since the derivative depends on the point x where it is evaluated, it is itself a function. Maple uses the notation D( f)\nfor this function; the more usual notation is f ' . You should by \+ now be familiar with several interpretations for f ' : " }}{PARA 0 "" 0 "" {TEXT -1 51 "It is the instantaneous rate of change of f at x; \+ " }}{PARA 0 "" 0 "" {TEXT -1 56 "It is the slope of the tangent line t o the graph of f; " }}{PARA 0 "" 0 "" {TEXT -1 85 "If f(x) represents the position of an object at time x, then f '(x) is its velocity.\n" }}{PARA 0 "" 0 "" {TEXT 258 32 "Maple's differentiation commands" } {TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 175 "It is easy to write a Maple pr ocedure which computes the derivative of a function using the definiti on. (The command unapply is used to convert an expression to a functi on.)\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "myD := proc(f) un apply(limit((f(x+h)-f(x))/h,h=0), x) end:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "f := x -> x^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "myD(f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "m yD(sin);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "However, the built-in command D does the same thing:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "D(s in);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "Note that D differentiat es a function, and that the derivative of a function is again a funct ion. For example, D(f) can be evaluated at a point:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "D(f)(3); D(f)(-5); D(f)(0);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "but Maple provides another comman d which is closer to ordinary mathematical practice (it is Maple's equ ivalent of the " }{OLE 1 3585 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N: F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::fyyyyya:nYf::wyyyqy;:::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::: :::::gjvb?f_nqCvr;V:>Z=B:<:=ja^GE=;::::::::: N;?R:yyyyyy:>:<::::::JDJ:j:VBYmp>HYLkNG>::::::::N:;::::::::_Z:vyyuy:>:<::::::AJ:n:>:nYN::wAJ=j=B:K:M :OJ:V;nYvY:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::=Z<>Z:B:::cR<:TnEj``pkDqqHqqTPtWdZfbk;>ZEZ?GHZhV@cYH_W V>Z::JW^EClqVFempfFWMtVH;kjj;^:all^HcmnnGIms^EemsfF_M::Vir:q?AB:s:;jZ:f:NZ;Z:f:FZ =f:V[v=>b\\:B:;xyyQVyyyyY:DZ:> m=j?^;UTRcETcTX[USJwEJ:><<:fb:Fh:fH=MtFGYMq>>Wlj:gmlB:;:::: :JJNZ:vyyuy:>Z:>Z::::::J`:J:<:::::::>=?R:AJ:^: vYxY:B::::::f::;:::::::::N;vYxI:;Z::::::JywYB:::::::::::: :yay=J:B::::::::^:f_;jb:f:V\\jxM:<:[V::[K<<:Uk:>;N`D^k=^D?J:JSng; vP=:CM?JMJ?vYxy:J:^=VY[j=>:;JXE::yayI:>:[K;B:MmlNH;Ka:;::::::::::::: ::FC::;::::::::JdfH_moFGaMnJ:::::::::CeW\\:<:n]yy]Z4:" }{TEXT -1 94 "n otation.) The command diff differentiates an expression, and gives ba ck another expression.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "diff(x^2, x); diff(x^3 + 5*x, x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 152 "Notice that when you use diff, you must tell Maple the indepen dent variable. Guess what the answer to the next command will be befo re you hit .\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "di ff(x^2,y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 516 "That's all there i s to differentiation in Maple. Just remember that the derivative of a function is another function, and the derivative of an expression is \+ another expression, and that they are computed with D and diff respect ively. This is a nice example of Maple forcing you to think clearly, \+ by the way. The distinction between differentiating functions and exp ressions was not invented by the Maple programmers: it is a real one, \+ and is reflected in the fact that we have both the prime (') notation \+ and the " }{OLE 1 3585 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyy y]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::fyyyyya:nYf::wyyyqy;::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::::::::gj tlPf_nqCvr;V:>Z=B:<:=ja^GE=;:::::::::N;?R:yy yyyy:>:<::::::JDJ:j:VBYmp>HYLkNG>::::::::N:;::::::::_Z:vyyuy:>:<::::::AJ:n:>:nYN::wAJ=j=B:K:M:OJ:V;n YvY::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::=Z<>Z:B::::::::::F:wyyAbR<:TnEj``pkDqqHqqTPtWdZfbk;>ZEZ?GHZhV@cYH_WV>Z:::::::V:C:;:::::::::JHJ ;vy;q=>:>pip:AR:=r:Ob<=r:WJ:@JCHRlR:JCHRvJ::vYxy;J:Vi:E:Mb:>Z:f:NZ;Z:f:FZ=f:V[v=>b\\:B:;xyyQVyyyyY:DZ:>m=j?^;UTRcETcTX[U S>r:>;Ny<>:[Z::ED:]E:emj^HMmqnG;KaFFJufF>?B:yyyxI:;B:;B:::::: ^:D_mlVH[KR<:;B:::::::JFNZ;V:;J:::::::::J?jysy:>:<::::::wqy[:::::::::::::vYxI:;Z::::::::Jl;B:;B:Cb:@CB:f?=J>JSdJ@IJdN:;B:CEXK:_Kq@jUG:^FO:G;Ojysy=:;JHjw?:sg:B:=b:?bBaTXaEWEUU>Z:N;;J@C:UK;^:>x;F:AZ:>::::WTJWTL>Z: vYxI:::::::::::::::::::::::::::::::::::::::::::2:" }{TEXT -1 85 "notat ion for differentiation, but it is usually glossed over or ignored in \+ textbooks." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 21 "Differentiation rules" } {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 264 "Now let's use D and di ff to compute some derivatives. While doing this, we will take the op portunity to verify three of the basic differentiation rules. (The re maining one is the Chain Rule, which we will explore in another worksh eet.) First, the derivative is " }{TEXT 260 8 "linear:\n" }}{PARA 0 " " 0 "" {TEXT -1 48 " " }{OLE 1 4609 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :fyyyyya:nYf::G:I:wAyA:::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::::::::gjtlPf_nqCvr;V:>ZAZ:j:vCSmlJ::::::::::OJ;@jyyyyyI:;Z:::: :::^<>:F:AlqfG[maNFO=;::::::::_J;@j:j<>:yayA:<::::::=J:nF>:V:Y:G:;:wA?:Jyyyyg:n:v:JyK?j?J@j@>:W:YJ :><:a:c:e:gJ:v<>=F=N=V=nYvY::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::F:DJ:i;V;;k\\Jt:>:::::=Jyyy;d:yayQZ HQ:R<:TnEj``pkDqqHqqTPtWdZfbk;>ZEZ?GHZhV@c YH_WV>Z:::::::V:C:;::vYxI:;Z::::::s:?jyAVoC:;:[xqgJHUUEB:;j;@j:HJ?dj:H JA>Z;N\\=`F@:N\\=@INZ<@?_rZ=M;\\S;hXCHRmNZ<:JCJ:B`N:\\C_jmJN:?bZM;gFDfB]E:emj^HMmqnG;KaFFJufF;J::::::>^:NZ :vyyuy:>:<::::::CZ:nZ;B:;B:N:YLpJbNHEms>@[C:>Z::::::::kJ;@j;>:C:yayA:< ::::::MZ:>o>Z:j::<::::::wqy[:::::::::::::vYx I:;Z::::::::JEa<<:E:cb:DJZjxM:<:[V:cTTUUSaEBW TSiEB_tUyyyyq;Z:j:jysy?B:j:F;;r:ac:^;UTRcETcTX[US>v:>;Ny<>:[Z::CZ:f_;juH f:V<^kJf:^<^]Qf:;jCJb]KHB:qA>Z:JBA:;b:;b:;B>aTXDpql`q?B:;JRJSNu=ntUF:>ys?;N@OW:gx@=:gYXK:_KJodP F:vvs?;N@oPeTk:JnaQ>JSNOgd?=:GYXK:_K:EJcLk:jO`Q>JS>N;v>=:iAN@oP:Ik:Jv; _KO?jMF:nXJS>N_b==B:vVJSdJgO\\HF:>WJSNO_b==:GAN@;N>lj:jX`Q>JSJwTj:jO^= VYZ:JB[:^Z@<:UK;^:>x;F:JSJWhk:jN`Q>JSJndk:Jv`Q>J SJfak:>:KSXK:_;SR@=:IUXK:_KPIJRXK;>Z:^fr?;;JSnKav?=:C=N@;NpPk:Jk\\Q>JS JlEk:>:MCXKB:N@N\\LF:>ir?;N`DJ;Ak:_r?;N@>sJF:nfr?;N@SOpuJ;Jl:_KAwc IF:^FJS>JWS==:=EXK:_;_gic>;N@>dCF:>_c>;N@^jBF:vfc>;N@ ^lAF:Nfc>;N@Nq?F:F?JSJnMj:Jv:_;kD;=:K;N@nm=F:nfc>;N@^p;N@>`:;JXE::[KX?B:AZ:>::::WTJWTL>Z:v:>;F[:>:O:Q:wAyA:::::::::: ::::::::::::::::::::::::::::::::2:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 20 "Second, we have the " }{TEXT 261 13 "prod uct rule:" }}{PARA 0 "" 0 "" {TEXT -1 49 " \+ " }{OLE 1 4609 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j:: >:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::fyyyyya:nYf::G:I:wAyA::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_ mvJ::::::::gjtlPf_nqCvr;V:>bAZ:j:vCSmlJ::::: :::::OJ;@jyyyyyI:;Z:::::::^<>:F:AlqfG[maNFO=;::::::::_J;@j:j<>:yayA:<: :::::=J:nF>:V:Y:G:;:wA?:Jyyyyg: n:v:JyK?j?J@j@>:W:YJ:><:a:c:e:gJ:v<>=F=N=V=^=nYvY:::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::j:b:;B:<::JRIj?J: vYxYyK:B::vYxY:B:jysy=Jyyy;d:yayQZHQ:>:;`:Z@wZ::::vYxY:B::j;^:;J::::::F:>::Z :AR:=r:Ob<=r:WJ:@JCHRlR:JCHRvJ;DRNN\\=pF ?bZJvNZ:_rZG=K;kM; 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:<::::::wqy[:::::::::::::vYxI:;Z::::::::J;a\\:NZP@?TsNB:=b:?vPDk;Y@ZrK ;a\\:yayQZ:J:jmq;Ir;nx?j:F;HjwEJ:^:s :;jZ:f:NZ;F:f:FZ=f:V[f@>^UN\\:B:;xyyQTyyyyYje[:^\\@j?<:G;Sj`@Pt\\Pd`QrP@[Q>>:oi:;JBB:J<>:US:f:< JDDJJLJZf:V\\<>^>^x;f:^<>x;J:JiOf:V<^_Q^:;jP>:JD>>OBl;J::[K<<:UK;>;N`DR;_i>=J:RXs:qQ:uI;B:>LB:Cb:^D :::G@=:;JRB:f?AJZ:F_W?;N@F^KF:;JvpO>JSjB=k:JNpO>J Sj\\yj:jmpO>JSjZxj:jgpO>JSnigQF:vFJS^n= ^kON:^fr?;;JSNl;vwOF:^FJSnIcC?=:MCXK:_;?u>=:kEXK:_;_h>=B:;JN\\Q>JSJFDk :Jm\\Q>JSJ_wHF:>?JSJHtj:<:IEXKB:N@NwFFZ:jN:_;kB==:k= N@>kEFZ:JN:_;Wh<=:G=N@COs]J;Jl:_KCgdCF:^FJS^O_S;=:M;N@>f>F:>IJSJbIj:JN :_;Cr:=:I=N@Wv:MS;=:M;N@vd>F:>IJSj]Ij:JN:_;Ur:=:G=N@SOdN:;Jl:_KAwP=:CM Hjw;<:[>DJ:DZJVdscRYEUXQZXAB:KB:N`D^nJS^N_G?=B:vvr?;N@_PSMk:Jn]Q>JSnO_G? =:GIXK:_KyN[KF:f_r?;N@CO=qj:ju;_KcojGF:>YJSnOGV==:gAN@COF\\j:jm;_Kc?]B F:>WJSnOkB<=:GAN@;F;kh;=J:far?[:JS^[?nw?F:vXJS>i>nw?F:vXJSNZ@nw?F:>YJS Nj=nw?F:nXJS^;Oc:=J:vVJS>IOc:=:IAN@?JODj:Jn;_K;O_N[:vYxy:J:^ =VY[j=:sg:B:=J;Dlc`qsLqlp@;B:OJ:^:>:AB: ::::WTJWTL>Z:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::6:" }}{PARA 0 "" 0 "" {TEXT -1 35 "The linearity rule is the s implest." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "diff(x^2, x); d iff(x^3, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "diff(7*x^2 \+ + 4*x^3, x); diff(5*x^2 - Pi*x^3, x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Here it is again, using D.\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "f := x -> cos(x^2); g := x -> x*exp(2*x);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "D(f); D(g);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "D(3*f - g);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 168 "If you look carefully at Maple's syntax, you can se e that this last answer verifies the rule, but it might be easier to c heck by evaluating the various functions at x:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "D(3*f - g)(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "3*D(f)(x) - D(g)(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Now let's look at the product and quotient rules, with t he same functions f and g. The product rule first:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "D(f*g);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "D(f*g)(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "D(f)*g + f*D(g);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "D (f)(x)*g(x) + f(x)*D(g)(x);" }{TEXT -1 0 "" }}}{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);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "diff(f(x), x)*g(x) + f(x)*d iff(g(x), x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "In 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);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "(diff(f(x), x)*g(x) - f(x)*diff(g(x ), x)) / (g(x))^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simpl ify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "(You can do the calcul ation with D if you wish.)\n" }}}}{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }