Detail solution
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Apply the product rule:
; to find :
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Apply the product rule:
; to find :
-
Apply the power rule: goes to
; to find :
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The derivative of sine is cosine:
The result is:
; to find :
-
Let .
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The derivative of cosine is negative sine:
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Then, apply the chain rule. Multiply by :
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The derivative of a constant times a function is the constant times the derivative of the function.
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Apply the power rule: goes to
So, the result is:
The result of the chain rule is:
The result is:
Now simplify:
The answer is:
The first derivative
[src]
(x*cos(x) + sin(x))*cos(2*x) - 2*x*sin(x)*sin(2*x)
$$- 2 x \sin{\left(x \right)} \sin{\left(2 x \right)} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(2 x \right)}$$
The second derivative
[src]
-((-2*cos(x) + x*sin(x))*cos(2*x) + 4*(x*cos(x) + sin(x))*sin(2*x) + 4*x*cos(2*x)*sin(x))
$$- (4 x \sin{\left(x \right)} \cos{\left(2 x \right)} + \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 4 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(2 x \right)})$$
The third derivative
[src]
-(3*sin(x) + x*cos(x))*cos(2*x) - 12*(x*cos(x) + sin(x))*cos(2*x) + 6*(-2*cos(x) + x*sin(x))*sin(2*x) + 8*x*sin(x)*sin(2*x)
$$8 x \sin{\left(x \right)} \sin{\left(2 x \right)} + 6 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin{\left(2 x \right)} - 12 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(2 x \right)} - \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \cos{\left(2 x \right)}$$
(-100*cos(x) + x*sin(x))*cos(2*x) + 200*(99*sin(x) + x*cos(x))*sin(2*x) + 19800*(-98*cos(x) + x*sin(x))*cos(2*x) + 1293600*(97*sin(x) + x*cos(x))*sin(2*x) + 62739600*(-96*cos(x) + x*sin(x))*cos(2*x) + 2409200640*(95*sin(x) + x*cos(x))*sin(2*x) + 76291353600*(-94*cos(x) + x*sin(x))*cos(2*x) + 2048967782400*(93*sin(x) + x*cos(x))*sin(2*x) + 47638500940800*(-92*cos(x) + x*sin(x))*cos(2*x) + 973942685900800*(91*sin(x) + x*cos(x))*sin(2*x) + 17725756883394560*(-90*cos(x) + x*sin(x))*cos(2*x) + 290057839910092800*(89*sin(x) + x*cos(x))*sin(2*x) + 4302524625333043200*(-88*cos(x) + x*sin(x))*cos(2*x) + 58249564158355046400*(87*sin(x) + x*cos(x))*sin(2*x) + 723958868825269862400*(-86*cos(x) + x*sin(x))*cos(2*x) + 8301395029196427755520*(85*sin(x) + x*cos(x))*sin(2*x) + 88202322185212044902400*(-84*cos(x) + x*sin(x))*cos(2*x) + 871646478065624914329600*(83*sin(x) + x*cos(x))*sin(2*x) + 8038517519938540876595200*(-82*cos(x) + x*sin(x))*cos(2*x) + 69385098593153721250611200*(81*sin(x) + x*cos(x))*sin(2*x) + 562019298604545142129950720*(-80*cos(x) + x*sin(x))*cos(2*x) + 4282051798891772511466291200*(79*sin(x) + x*cos(x))*sin(2*x) + 30752917464768184400530636800*(-78*cos(x) + x*sin(x))*cos(2*x) + 208585005413210294194903449600*(77*sin(x) + x*cos(x))*sin(2*x) + 1338420451401432721083963801600*(-76*cos(x) + x*sin(x))*cos(2*x) + 8137596344520710944190499913728*(75*sin(x) + x*cos(x))*sin(2*x) + 46947671218388716985714422579200*(-74*cos(x) + x*sin(x))*cos(2*x) + 63382530011411470074835160268800*(x*cos(x) + sin(x))*sin(2*x) + 257342790382278893106879057100800*(73*sin(x) + x*cos(x))*sin(2*x) + 1341858835564739942628726512025600*(-72*cos(x) + x*sin(x))*cos(2*x) + 1568717617782433884352170216652800*(-2*cos(x) + x*sin(x))*cos(2*x) + 6663023183493881094432297163161600*(71*sin(x) + x*cos(x))*sin(2*x) + 25622387757113086777752113538662400*(3*sin(x) + x*cos(x))*sin(2*x) + 31538309735204370513646206572298240*(-70*cos(x) + x*sin(x))*cos(2*x) + 142431076223503608771305449036185600*(69*sin(x) + x*cos(x))*sin(2*x) + 310671451554996177180244376656281600*(-4*cos(x) + x*sin(x))*cos(2*x) + 614234016213859312826254748968550400*(-68*cos(x) + x*sin(x))*cos(2*x) + 2531388672881359592253655935143116800*(67*sin(x) + x*cos(x))*sin(2*x) + 2982445934927963300930346015900303360*(5*sin(x) + x*cos(x))*sin(2*x) + 9976649475473593687117349862034636800*(-66*cos(x) + x*sin(x))*cos(2*x) + 23611030318179709465698572625877401600*(-6*cos(x) + x*sin(x))*cos(2*x) + 37626220878928981905699719479673487360*(65*sin(x) + x*cos(x))*sin(2*x) + 135872464285021323548360098121043148800*(-64*cos(x) + x*sin(x))*cos(2*x) + 158531203564920906412547559059462553600*(7*sin(x) + x*cos(x))*sin(2*x) + 470045281850884578761894393499824947200*(63*sin(x) + x*cos(x))*sin(2*x) + 921462620721102768522932687033126092800*(-8*cos(x) + x*sin(x))*cos(2*x) + 1558571197716090971684176146867840614400*(-62*cos(x) + x*sin(x))*cos(2*x) + 4709697839241191928006100400391533363200*(9*sin(x) + x*cos(x))*sin(2*x) + 4955457141456289243303534415682365030400*(61*sin(x) + x*cos(x))*sin(2*x) + 15114144281441682192075779967831213342720*(-60*cos(x) + x*sin(x))*cos(2*x) + 21429125168547423272427756821781476802560*(-10*cos(x) + x*sin(x))*cos(2*x) + 44236519848121996659733990149749892710400*(59*sin(x) + x*cos(x))*sin(2*x) + 87664602962239458841749914270924223283200*(11*sin(x) + x*cos(x))*sin(2*x) + 124283555763771323948776448515963984281600*(-58*cos(x) + x*sin(x))*cos(2*x) + 325089569318304659871489265421343994675200*(-12*cos(x) + x*sin(x))*cos(2*x) + 335276569037150548326931814601205166899200*(57*sin(x) + x*cos(x))*sin(2*x) + 868671110687162784301596065103122477875200*(-56*cos(x) + x*sin(x))*cos(2*x) + 1100303157692723464180425206041471981977600*(13*sin(x) + x*cos(x))*sin(2*x) + 2162025875488049596483972428701104833822720*(55*sin(x) + x*cos(x))*sin(2*x) + 3418799097116676477989178318771716515430400*(-14*cos(x) + x*sin(x))*cos(2*x) + 5170061876167075122026890590372207211315200*(-54*cos(x) + x*sin(x))*cos(2*x) + 9800557411734472570235644513812254010900480*(15*sin(x) + x*cos(x))*sin(2*x) + 11880142183532853471891578377876561251532800*(53*sin(x) + x*cos(x))*sin(2*x) + 26032730624919692764688430739813799716454400*(-16*cos(x) + x*sin(x))*cos(2*x) + 26235313988635051417093902251144072763801600*(-52*cos(x) + x*sin(x))*cos(2*x) + 55683115404449905048525833349367011580313600*(51*sin(x) + x*cos(x))*sin(2*x) + 64316158014507476242171417121892916946534400*(17*sin(x) + x*cos(x))*sin(2*x) + 113593555425077806298992700032708703623839744*(-50*cos(x) + x*sin(x))*cos(2*x) + 148284475422336681336117433919919780737843200*(-18*cos(x) + x*sin(x))*cos(2*x) + 222732461617799620194103333397468046321254400*(49*sin(x) + x*cos(x))*sin(2*x) + 319982289069252838672674462669300579486924800*(19*sin(x) + x*cos(x))*sin(2*x) + 419765023818160822673502436018305164220825600*(-48*cos(x) + x*sin(x))*cos(2*x) + 647964135365236998312165786905333673461022720*(-20*cos(x) + x*sin(x))*cos(2*x) + 760329099746102622201061016184099920098099200*(47*sin(x) + x*cos(x))*sin(2*x) + 1234217400695689520594601498867302235163852800*(21*sin(x) + x*cos(x))*sin(2*x) + 1323535840298771231238883991135285046096691200*(-46*cos(x) + x*sin(x))*cos(2*x) + 2213914496499762786799587766989931349834465280*(45*sin(x) + x*cos(x))*sin(2*x) + 2215981242158169821067579963875383558589644800*(-22*cos(x) + x*sin(x))*cos(2*x) + 3558076869374618764499337482662389669376819200*(-44*cos(x) + x*sin(x))*cos(2*x) + 3757533410616027087897200808310432990652006400*(23*sin(x) + x*cos(x))*sin(2*x) + 5493171307104674583788450850426145454476492800*(43*sin(x) + x*cos(x))*sin(2*x) + 6027709846196543453501759629997986255837593600*(-24*cos(x) + x*sin(x))*cos(2*x) + 8145047110534517486307013329942215673878937600*(-42*cos(x) + x*sin(x))*cos(2*x) + 9162118966218746049322674637596939108873142272*(25*sin(x) + x*cos(x))*sin(2*x) + 11596338259066092692369307113816035874675097600*(41*sin(x) + x*cos(x))*sin(2*x) + 13214594662815499109600011496534046791643955200*(-26*cos(x) + x*sin(x))*cos(2*x) + 15848328954056993346238053055548582362055966720*(-40*cos(x) + x*sin(x))*cos(2*x) + 18108888982376795076118534273028138195956531200*(27*sin(x) + x*cos(x))*sin(2*x) + 20784693710238679798344987613834206376466841600*(39*sin(x) + x*cos(x))*sin(2*x) + 23606230280598322152797375034483108719729049600*(-28*cos(x) + x*sin(x))*cos(2*x) + 26148485635461564907595306998049485441361510400*(-38*cos(x) + x*sin(x))*cos(2*x) + 29304285865570330948300189697979031514146406400*(29*sin(x) + x*cos(x))*sin(2*x) + 31544204893572681475829259235742236405451980800*(37*sin(x) + x*cos(x))*sin(2*x) + 34676738274258224955488557809275187291739914240*(-30*cos(x) + x*sin(x))*cos(2*x) + 36472986908193412956427580991326960843803852800*(-36*cos(x) + x*sin(x))*cos(2*x) + 39151156116097995917487081397568759845512806400*(31*sin(x) + x*cos(x))*sin(2*x) + 40400847036768088197889012790392941242367344640*(35*sin(x) + x*cos(x))*sin(2*x) + 42209840187668151848540759631753819208443494400*(-32*cos(x) + x*sin(x))*cos(2*x) + 42849383220814638997761074171628877075238092800*(-34*cos(x) + x*sin(x))*cos(2*x) + 43488926253961126146981388711503934942032691200*(33*sin(x) + x*cos(x))*sin(2*x) + 1267650600228229401496703205376*x*cos(2*x)*sin(x)
$$1267650600228229401496703205376 x \sin{\left(x \right)} \cos{\left(2 x \right)} + \left(x \sin{\left(x \right)} - 100 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 19800 \left(x \sin{\left(x \right)} - 98 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 62739600 \left(x \sin{\left(x \right)} - 96 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 76291353600 \left(x \sin{\left(x \right)} - 94 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 47638500940800 \left(x \sin{\left(x \right)} - 92 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 17725756883394560 \left(x \sin{\left(x \right)} - 90 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 4302524625333043200 \left(x \sin{\left(x \right)} - 88 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 723958868825269862400 \left(x \sin{\left(x \right)} - 86 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 88202322185212044902400 \left(x \sin{\left(x \right)} - 84 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 8038517519938540876595200 \left(x \sin{\left(x \right)} - 82 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 562019298604545142129950720 \left(x \sin{\left(x \right)} - 80 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 30752917464768184400530636800 \left(x \sin{\left(x \right)} - 78 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 1338420451401432721083963801600 \left(x \sin{\left(x \right)} - 76 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 46947671218388716985714422579200 \left(x \sin{\left(x \right)} - 74 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 1341858835564739942628726512025600 \left(x \sin{\left(x \right)} - 72 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 31538309735204370513646206572298240 \left(x \sin{\left(x \right)} - 70 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 614234016213859312826254748968550400 \left(x \sin{\left(x \right)} - 68 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 9976649475473593687117349862034636800 \left(x \sin{\left(x \right)} - 66 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 135872464285021323548360098121043148800 \left(x \sin{\left(x \right)} - 64 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 1558571197716090971684176146867840614400 \left(x \sin{\left(x \right)} - 62 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 15114144281441682192075779967831213342720 \left(x \sin{\left(x \right)} - 60 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 124283555763771323948776448515963984281600 \left(x \sin{\left(x \right)} - 58 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 868671110687162784301596065103122477875200 \left(x \sin{\left(x \right)} - 56 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 5170061876167075122026890590372207211315200 \left(x \sin{\left(x \right)} - 54 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 26235313988635051417093902251144072763801600 \left(x \sin{\left(x \right)} - 52 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 113593555425077806298992700032708703623839744 \left(x \sin{\left(x \right)} - 50 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 419765023818160822673502436018305164220825600 \left(x \sin{\left(x \right)} - 48 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 1323535840298771231238883991135285046096691200 \left(x \sin{\left(x \right)} - 46 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 3558076869374618764499337482662389669376819200 \left(x \sin{\left(x \right)} - 44 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 8145047110534517486307013329942215673878937600 \left(x \sin{\left(x \right)} - 42 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 15848328954056993346238053055548582362055966720 \left(x \sin{\left(x \right)} - 40 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 26148485635461564907595306998049485441361510400 \left(x \sin{\left(x \right)} - 38 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 36472986908193412956427580991326960843803852800 \left(x \sin{\left(x \right)} - 36 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 42849383220814638997761074171628877075238092800 \left(x \sin{\left(x \right)} - 34 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 42209840187668151848540759631753819208443494400 \left(x \sin{\left(x \right)} - 32 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 34676738274258224955488557809275187291739914240 \left(x \sin{\left(x \right)} - 30 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 23606230280598322152797375034483108719729049600 \left(x \sin{\left(x \right)} - 28 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 13214594662815499109600011496534046791643955200 \left(x \sin{\left(x \right)} - 26 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 6027709846196543453501759629997986255837593600 \left(x \sin{\left(x \right)} - 24 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 2215981242158169821067579963875383558589644800 \left(x \sin{\left(x \right)} - 22 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 647964135365236998312165786905333673461022720 \left(x \sin{\left(x \right)} - 20 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 148284475422336681336117433919919780737843200 \left(x \sin{\left(x \right)} - 18 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 26032730624919692764688430739813799716454400 \left(x \sin{\left(x \right)} - 16 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 3418799097116676477989178318771716515430400 \left(x \sin{\left(x \right)} - 14 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 325089569318304659871489265421343994675200 \left(x \sin{\left(x \right)} - 12 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 21429125168547423272427756821781476802560 \left(x \sin{\left(x \right)} - 10 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 921462620721102768522932687033126092800 \left(x \sin{\left(x \right)} - 8 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 23611030318179709465698572625877401600 \left(x \sin{\left(x \right)} - 6 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 310671451554996177180244376656281600 \left(x \sin{\left(x \right)} - 4 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 1568717617782433884352170216652800 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \cos{\left(2 x \right)} + 63382530011411470074835160268800 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 25622387757113086777752113538662400 \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 2982445934927963300930346015900303360 \left(x \cos{\left(x \right)} + 5 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 158531203564920906412547559059462553600 \left(x \cos{\left(x \right)} + 7 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 4709697839241191928006100400391533363200 \left(x \cos{\left(x \right)} + 9 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 87664602962239458841749914270924223283200 \left(x \cos{\left(x \right)} + 11 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 1100303157692723464180425206041471981977600 \left(x \cos{\left(x \right)} + 13 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 9800557411734472570235644513812254010900480 \left(x \cos{\left(x \right)} + 15 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 64316158014507476242171417121892916946534400 \left(x \cos{\left(x \right)} + 17 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 319982289069252838672674462669300579486924800 \left(x \cos{\left(x \right)} + 19 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 1234217400695689520594601498867302235163852800 \left(x \cos{\left(x \right)} + 21 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 3757533410616027087897200808310432990652006400 \left(x \cos{\left(x \right)} + 23 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 9162118966218746049322674637596939108873142272 \left(x \cos{\left(x \right)} + 25 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 18108888982376795076118534273028138195956531200 \left(x \cos{\left(x \right)} + 27 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 29304285865570330948300189697979031514146406400 \left(x \cos{\left(x \right)} + 29 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 39151156116097995917487081397568759845512806400 \left(x \cos{\left(x \right)} + 31 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 43488926253961126146981388711503934942032691200 \left(x \cos{\left(x \right)} + 33 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 40400847036768088197889012790392941242367344640 \left(x \cos{\left(x \right)} + 35 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 31544204893572681475829259235742236405451980800 \left(x \cos{\left(x \right)} + 37 \sin{\left(x \right)}\right) \sin{\left(2 x 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\sin{\left(2 x \right)} + 8301395029196427755520 \left(x \cos{\left(x \right)} + 85 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 58249564158355046400 \left(x \cos{\left(x \right)} + 87 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 290057839910092800 \left(x \cos{\left(x \right)} + 89 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 973942685900800 \left(x \cos{\left(x \right)} + 91 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 2048967782400 \left(x \cos{\left(x \right)} + 93 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 2409200640 \left(x \cos{\left(x \right)} + 95 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 1293600 \left(x \cos{\left(x \right)} + 97 \sin{\left(x \right)}\right) \sin{\left(2 x \right)} + 200 \left(x \cos{\left(x \right)} + 99 \sin{\left(x \right)}\right) \sin{\left(2 x \right)}$$