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t1lo7/2025". if ys Soe : [vs Ay = £ Gee) —O y=> dependent va ; @-@ x“ indegendenk varishe ; y =f Gt 8a) -F Od => by = £6)1 3x-fG) 7 | dx Bx | Yim sy = im $Go4 de - Fd [2x79 Sy x0 5 : | dy. = F'GO = lim _FGs x) LOO ---> First Order I dx 3X0 ax frinci gle | _ __ | ats y = Si 2 | Find dy by Fisk osder princiale, | dx => | dy - lim _£ Gard FOO. | dx 9 yx | = _\im fis Ge BO gin a \ i 3X0 i" Sx J | = lim 2 gn x4 Bv-w as Gr Bea x) | x0 2 2 ) 3 \im > (se i) (9g GX>0 = Bx 5 lino (sin = -\ ef ie Doe \ re w > wo lL ae 2 yy Cos x fora] g 21x soe? FO a eee a ae ax dx al dG”: ix Dd Got x) = ~coser*x 7 _ ax dx ac _d Ce”) = - Mt) id Conget bm crete 2 Geb oe —_ __| de dx id Clog 20) = 1 ait) A Gite) = dx Ps Casi dx (ax? 5) | A GD = olga 2) d Coc) = = 4 dx ; \-x* 6) : Coin x) = cas ®d Cok. 4 | dx dx ; Anat D | ad Cosx) = ~sin x 4) d (Ceosec™*x) - — 4 | dx dx (xl Joo 4. 8) | d Catt. a]: 5 ‘\ : 16) 4 Coc) = 1 [dx ad dx al fe ] | . “Algebra. of Differentiation ; oie @ | | Lene k£, @ Bd ko GO tot kn fn @ kis Gretant LG) = kt’ OF: Kf," Gs. wt kata “GO ail ys 2 sin= eee” es po - dy 2 ost-ee de ; dx 2 I2x | 2 ~&y: fade x4 242 - | BGO. a°s 32 1 f£@-= a GOS ___ | F'OO2 2- | Foose Ixs tee ~ ae _ i fGo. 1-4 @* _ | 2x 2 Faye 4 - zZ << xe chain Rates r | =f Fs £00 ae Ge) it ae a) a) V0 ea ——~ [de du] de / mz Vers | ag Te _——— _dy. = Ce ). i - e [f. fad w= 0G) ve v@ % \ 2h zie _df . d&- dua dy ' de du _ dvi dx | | : | ye sin'x * ye Sndx - | | dy = d Cont) A Gin) ra d (sin Ja) . PRO@ED) | da d Csin x) dx dq) dx i $= 2 sinx @sx = sin 2 2 i Cok dx) : j dk Gh) 2 i y = Sin AX . ys = sin Coos x) | dy. = d Gein? dz) . d Cae "dy = d Coin Cons ®)- i Gost doe 4 Gin x) A® d Gesx) dx | dy = 2 sin Ix os Ix i Cae Cos i) er x) { ae Tk | dy = _ Sin AX OS tx. ot ik ye. ta. * Gin x) wd aig? | | —_— ae 109," Gin C si : Csinx) | dx d Ct ls, sina). A Csin x) .. dx . dy = 2 Log Cain x) { 1) Gs X i { x \sinx a | | dy = Cote -Csin x) Cot x) | dx Le - | | _ I —4 Ton Ale) = 02x -_ dG) - _ | + y= log,” ere) —> |S Kee’. — ~—# dx = | ia, + yn 24. > | = = f SF 7 ™~ dy — d Clog’ Gin? Ged) Ja) u ad Clog, Gin* Garg) AGE 7 | dx j d Clog. Csin® Gud) d “Gan? Gass) 4 Ging) | -_d Cin ComtD) « d Cou3d / "Ca 7 \ Gates) | dy = 3. log” sin’ Qx+3) 4 2sin Ox2)) Los dx \ sin? Qua) J. ‘ pee | y= loa. Gus Jods 2 _) . | dy . d log Gus dx*a” ) d Ge Per . J de due [ates ercrol | dy. (— L a + 1. Gx) ) 6 ita pd Nedac)/ ater / | dy = (— 4 l\ (44 x A - Ae) J . Gulia) Tea] Pere | = — | Cox+b) = 0 —>: d Gee Cxsh)) I dx dx = d Cas Caxsb)) =—O cas « Gna) O_sec Cax+b) ton (oust ma => d CG uy — | d Coe Cons) so oan Cort). dx dx " Gout » | _d Chon Comb) = a soe Gans i av Sarah Gs wee oe ax46 | > | t = =0 coer’ Could dx am cosec Caxsb)) — ~a cosec Cansh) cat Coxsb) dx = OT) es cir [ENBRe E | foe f/f a Ly: C3 TT dps 43 = \O +- 2 + ten du 2 (Ta ey 1c ‘re 2 Chas) 2x43 yi —mem:| ye Clyaa- 4) | ’ 42K43 y= DH43 + 1-2 | 2n43 | dys Ds 2 | dx (2_43) : | dy. 2 2x43 — ied 4 12.) | de. fixe3 / be 3 Qx+3) 2443 y] | = 9 if tot & \eera of be 40 \ CO oe3 ) Gora 203] | = of 4 i \ iW Cx13y ) Pare 2 Ceci tye 7 en aa 3 ost Gog Ca, Gas 1y¥) Ghd 4) Cie dy — d ie bl “Cs oa 2 0! an i . 7 dd Coos®Chark 3) d Cos Gas) d Cam d Cho! | : Ly | y= log 9 09 09, Cog, 2) i dy la Ae Gren va i) | de eg. 4 los 5 (loys) / | 10303 log, x / Gk x_/ | ys tog tan {rs x TAT | dy = 4 sec. f+ 2) -L = fan (Gaes rs — 7 24 2) | Ly + + ec T+ fete Ges) ; cog (Ee ST 4 7 | 5in é= 2) Cos ed) : {i [ ~ 7 — a / f 7, -_ Higher “Onder Pieiadices. -_ ” Z ; i a ys POD — | dy < Gis 4 — —— | dx a — Td Cay). dy — 8" Gs 2 ed Lb We ae he . — | : | y= f@) ae : y= sin Cox+b) | dy = F/O) —> 1" order dy = 4 w¢ Caxsb) | axe dy. dy = @)—> 2" order dey . -a' sin Cax+b) do * * de® d*y FG) —» 3 order dy. = -o Gs Cart) du? rhe yrs x thas o_Lhen y ndty na Bins en of “x only x — a function of Sy’ onl/y O fancten of xt ky 2: —_— yrs On+ bate 7 Differentiating both sides tart “2 2y dy - 2ox+b—-@ "de Multelying both sides by Again differentaking both < ark de pra ee qT dx Ax 2 (dy mara) = 2a => y? Bip eee y dy are 9) xe = M Cox thx 4c) a (3 2ax+ 2 2 = Oa + abx+ ac ~ Cons 22 =O X+ abxt+ac- ax = abs “= /QC- G a SSS —— | feo = Gis) Ca &) Cosa) | ~ Cant gees hoy, GD» Tag, Coes 19") Coe) * Gonead = su) Teas — log FOO « 109. 2D) 3 3 2 q Ne os lage. Rx’t 9 3 2 8 a =4 log, Cen 4 4) —~ | Diferentiading beth sides web “n’ : | i g 2 3 flan» 2 ( mm \, 1 3x, 3 Cond-d tx 4 Fo) 3 Net J 2 we won 8 aute 28) fm . o| f@=-_@ | L@=o IO) @"*@) Heroegenesus Equation *r : xy = Guy) then dy ¥ dx wm _— log, Gory") 2 log. Gary)" log. + Log, y = Cran) loge Gay) m Clog, 2) +h Cog, > = Cmin) tog Gury) Differentaaking beth sides art “2! my pdp = Gun 4 (te dy) x Y dx Wy m4 0. dy = Min, _ meh dy xm Y dx xy dey de ae men_\ dy. Mth — mM \Y xy] doc Ary wx on = = = my \ y Coceg 5 de x Cory? Yn — = - = = 2 dx . wx doe es — ca = ae»| = — J olution: fl y = Ox ae + HC =) 2 Gee) = i ene Snee eN 2 1 Gea) Geb Gs) GR) OD EO yee, ox. BA. bt x abe 7 Ga) Gcb) Ge) Orb) Cee) . a it f Ged ele 2" GH) Gd —> i : vi! f aa Gb) Go) ey : | ! 7 bh) - Fegtata = _ a ees) (Cxe-b) (-¢) | tog, f= log, x log. Ge a) Geb) G-¢) ! y = 2 Is. x = leg, Gead- Jog G- b)= og, Gee) || Differenbiaking beth. sides hy So xe! [ 1 od 2 3 je eee ob LY dy % X-0 Xb KC i Lx x-a J me xb)” Cx xe J py. fis ais fois \ fA oe eee cs = Lg [F"GO- 20 _constant ‘pr ‘order_ derivative [2h pa F tak [coe logy. ie a — tas x SCF) CEs Pi — fa = = (waar) y= ees —_ —_d+Gqo) .0 aes _ . | £@= xt f° OO = axe ome = dxf ay faramettic. Differentiation I % = FC4) ye = £649 9G) _ oe ee = = d Csnx) (dx) | dy cide \dy / d Ginx) = Cosx) de a —~ | dy dy — |] Tg | a GD = d GA). dt A Gay). AG | doe. dt dx dy | — | a G2 Ge) de iGpeun dy | dx do dx dy | _ “iution: ary = t— = be Aiky 4°, ay” = = + —_ —Difierentisking \eth cide ark ‘x’ Cosy?) re (4. a 2a y* ex “ly dy fe) atays + 2x wy Hs "aE = 3 “dx cba» 2x y* ne ny? 2 ay. ay dy iS 2x y= «2. dow 3 ey es -4 xy dy = — xy aaa "dx , xy dy = =x ys 4 do © || Differentiation ot G) -| we £@ NEMEC) du = £’/@) dy = g' Cx) dx dx du. dee dv eo Ses1-l _ Ze 4 cores {-y* = ati y — | mw: Gsm, y= cos f a | {4 wen + S4- os 8 = a Coos x- cos P) | Sin + sin B= 0 Cos «= os PB) 27 Sin A+B cos K-8B = +26 sin X48 gn K-8 | 2 2 2 2 — | . Wi | Cos 7. =—=0_Sin a "@$ x=05 y = 2at"*Ca) | os => _-i ‘; 4 dy =O _= cos =~ =a =a {-%¢ Ti dx sin > _L dy Pa => | wt “FF =-a 4-y? dx 4-2 2 dy _ ¥ => | KP = 2 a@t7* Ca) dx reas ree. X Scluton:|| © = s4e4 - Ji-t x= log toy -6 rt + 14-4 ° 4 x = log. doe - = -1 sec (T-€ atk + (Hat tae) 4-4 de tan (te) <0 fut = @¢ 28 pee = 1 hae . f+ ms2e — 1-os28.) de race avers + 4 1-@s 28 doe wie = x leg, (—. Des? — 4 2sin*O de 23m Ch-s) @s ex rr © + 4 2sin26 = -2 a a | n= {og [Sees - [2 cine sin 2 Ge Sin (#-26 5 os 6 + {2 sino =. Paes = = log, Fee -sgne_\ Os 26 + cs 6+ sin 8 dey = ee = lege (A= tone) y= 24am ( |2ant® \ “t+ tone / ‘ \ [2a.*e / — =-i]|_dy -2 dy = 2 dy =-4 du 2] | de dx = dx i ax™+ 2hxy + by*= nd ae : _ Ss —~ sludion: yx cs Cees x) ee —_ { Casta Wiig oN hal I ae 1-os?x [sin 2 oo | | f£@-<=_ sin x p i [sin x| ~ Lf CE) = sinx - —~ | -6in x ace. | __— Solution: | y= a 1. | y _ ay +4 . | x a a | Difertntialing beth sider wk) [ Hib... | 2y_dy 2 2ay +20 dy [ y el pty I dx dit: - | | ys xy + 7 0% [ Coy=2°) dy = 2ny | y | dee [ : ) | > dy =o 2ny | [__ [de oy — Sy | y= sina Sein + gin ed ns = | ys sinx+y | (Cry. 0D) dy = 5% Differentiating M cides wr - : | ie _2y—- /. de ar | Yo @sx+d | 2y d Qny xd | dx 2 Sin x+y ths Vy) | Y . SR de | dy = Sh SF eee dy é [ Gy-2) dy = 2xy | dx__2y dx_/ | _ dk , [dy = 0s X 4 A dy j | dy _ poral [dx ty | dy dx dx "2dy-4 [ G - 1) dy = osx X 2y Z doe 2y | iis. | =L77~) a — 2 a gg -y = a “a oo Clog, x) + | + =o SN ~—Siubon:_Diffrentiaing_be_s re i -dy = -a ine i dz Xx 4 a | .. Lx dy =~ sin Clgx) + b cos Clog, x) a = 1 ae _ | Again different aking beth cides tark 20 | dy 4 x_ dy x "=a os Cg.x) bs 2 Clog. x) | doe dx? Hw x J | eg: y= a_sin x + b cos x pen d?y +y =0 | Pi ude ion y= Asinxth asx | dy = 4 @sx-b cnx ne ae LL” n i} i b> In, S ad Le lb ie a ad NZ r | <= —~ - ge _secQ- ws © ky = c"e an =_Sec = os” © , then CAS a “a4 +43 sec - ase = ae sxc8 tone + sine =lma (sca + sna. _ : | dy = tone Csece + ose) de ae | h h LY sec. & = os 6 “a ty =hcec” © Coro bno)t nos 8-ane 8 —_ dy = nsec"6 te + bh os © sne \ Te eer: |e dd A hr ( _—t 4 1 C. © © _ \ Sec 8 + Os e) ri _ nine Ceec"e tase ) ' Gece ase) dy = nGecere ___—_- a8 H dhe Cec 4 0% e) (dy \ =h. Cece” © + os re) Kd) C Cee a= oe @) (ap) n (Gece ~ cos oe 4) “A G2rg) (— _dy \* = n Gite) Xe Es. ; akon = ax’ + 2hxy + bys a | di eee Corh “Be Gouaby) Differentiating bah sides cork 2” me Cha by) 2ax + Ih (y+ dy \ + by _dy =6 de / du. Coons 2 hy) + Che + 2by) dy <0 dx — dy = = Caxshy = dx at by) 
