1. CIRCLES 2. DIFFERENTIALCALCULUS I. LIMITS II. DIFFERENTIATION III. SUCCESSIVE DIFFERENTIATION IV. LEIBNITZ’S THEOREM V. PARTIAL DIFFERENTIATION VI. MOST COMMON TRIGONOMETRIC FUNCTIONS IN THE DIFFERENTIATION

1. trgle b Tce lines' o à ary hleo the hoo lines
a2ty+e,
=o y+S=o.
hen
aya+bb
he lre axthyte
b J t = o areparallel
and ey e-
hebines a thyto
Ci tb+G:o areer
L/
and y

2. Cincles
A eizdi s he set q all point in a plane hatas
uicls
tunt Jrom a given potnt
caled hocente ef
the cince
h e distane demo ss a cincl hnough cnter d the
cibde is Caled elia metez
>he destance from centee eem of he cizcle to the
Dg poin on he cizcle is aled eadius.
The line sègment that jorins too poin on
the pirh
t to ahor s belore the diamefr iy knern a s
chod
O>center DE£FG->cherds.
oA> Radius
D CoB->Diemtes
Ciircl
Nete Evey
d i a m e r r is a chomd but every
chord s nof
d i a m e t t e s e .
*Equation ef he eircle eoih cenkrs a l Cuf) &
Pealtcd
- r ) +C-J= a*
eP =

3. + 2gx
+
2fy +c o epescnfs cicle eguaim
centree) : Ca,{)
Raelds() +4-e
Nete: Anade
genezal
steond degnee eguahn
s
ax tahty ty* +2gr tfy t c a c
eepaesenk a cicle
asb h=o t-ac 2o
center A)
a e
radiu
Note f he cizelk ty +29 tafy te =
o pa gi
hrangh gin Co. o) then Cao
Note he centhe q he cince r ty*+292 tay te =o
ies on -ax hen tentre
+ 2 9 x tc=o
Netep He cente the einhe 'ty t24r t y +
lies o n - a x i s
hen cen te CO, 7.
Note: A ciecle of asodius 1 is caledenit cizde

4. Not two tz mota eizche have same centes hen
hey arze Said o be eoncentade eizeles.
And he egua tion U concentreic circle aa ný
dyfere CeNn tants.
h e eguationqhe
e i p e k d h o s e endsq
diameter Az),B(y), Ben h- guahn q
cinelk
*oameic cguafion ecic
at the pasamefric egaation q
eiacle asoih cerwe
Cy) 2adius are
, SinE.
whereo' bpasra metr O<3 < 9
Nete he kngh -itercptmade by the einele
9 t9 teo
-
Noe( 7he kngth dA-nterceptmad
hy he cisc

5. Note p a cecl t * +29 t3tyteo touches
a z i s
touehs -0z ( <)
touehes oth a r B EJ-Oxs tf

6. Differential Caleuleus
imits
Liri a funeicn let )
be a funehon
&a' beba
eal Dumbez
uhen e v e give
vafucs o
chich
a u n e a n n e
&ncaie o a fa) taks values
n e a r e z
g veazs oa
e a l ncomber 1, hen c v e say
hat fl>) pproaches las
hat fl) appaoaches las
p a o a c h e s
a, o bimu 4 Fz) az fenols fo a àl
P denokel . fex)=l
Lt
a fCz)=/
lmit points: J ) à a funefin
z hen
fla)is
Called he volue o f t at a .
Right limt:
eppreaches faom ight to'a h value
to 2aa
'
the 2ght
binit f:incarer
to a a
2
> 2 S2)) a
e t lim: x
epproaches
Jramlet
o a' he veler-
s
the lptbmit ef
fineaer
o 2 t a
2
2 J o Y x - - a )
Limi: X riaht g 2pt limit exiat g a egaal then hr fati
imat ef funchbn 4C
e r i s . he emmnmulur
Caleed
he Limit 4 {lo)
i e .
imitflx) lifl) =limitfex)
>a.
a nay eproabes from kp/ ght ' a

7. Basie preepeahes linik
f x ) = l ,
Lt dr)=m
t h e n t t ) x ) = t n .
Lt fx)-, ceR hen 4)ca)=cl
L fCa)=l z)=m hn Ly))=Un.
f(x)-1,+.
J)=m then lit t)
a
a )
f a ) Egz à bouneledher atJ
Lt
fCr) -go) henSde)=2- g
f r ) =too =>
+R)
f fr)=othen fCx)
f Cf- )
then
4 ( ) a polynemvial funehin then
LE fCx)fCa)
a
"Note: f x - a m c a n s i thot x #a,
ifmean4
the 'e value may
he greefes saks Men a'.

8. ->ndehsminek Joms
n he pmoceds Lindisg He valueA ef limits,someime
twe opain he Jolowiy Jorma keO, Oxao,oo -
o o chih a s n nof dleterine ese foms a r
knoaun a
h v e e r m n a h fom.
Standasmd mits - a
- = n.an-
e
Lt
n 2eal umbes hen
Z , a M - )
m,n e a r roo ealnembeis he
- a n
tan I
Lt Sin
LE
Lr
LE CI+)
2 Deerentiatio:
-eonstant) =O
d
C =n.x-
d

9. ( e ) e
dr
(a)=a loq,
dx
d
(ley)=
-Ccosr) = -Sinx
( S i n z = < o s e
Ctanx) zsec?z
(seez) = Sec taN
dx
CCesec:) tsc farE -
Cbsee3 Cotx
d
Ccotr) =- Cosec1x

10. eperhes of diffesenta! function
d d
d r
d
d
Dferentialion ed compostk
funcionschain Ryle).
dCe . (M) xd6)
CF Comprsik Funetron
MF Main Funehon
SF Subs Funehon
then
Nole' Y u a functin
in z&g è a funcion in u
CP)MF) (GF)
d . du , iti knana
chain rcale.
d du
L4 y =
12)
hen d(ax+ay?=
dr
(M-F) CSF)
=nCaxa) d Cart2)
nCaz +2)". 3.

11. Dertivotire f Multpticotion tuv fune tions
the
f()&
gcx) a2e any
tusv
forem, then,
d e
dr
u ) ur +vul
dr
( u v w )
= uvw + vwu + ua
Divisin Rule
f +O) Eglr) are any
two funefrions. hen
d t a J ) o ] -
to)lge7
dzgC),
C)
Vu'-Uv
dd
u > s a den' vatve f u
a deAVatie

12. Legaihmie DYfeeentiation
Step-
hea iven funcFion id
:
Step -
i a Junchon f X.
Then takig
Both sides
* Cemmay
u s e d
fosmala
loqm = mleja
Sfep ~G echain rule, hhh dyerthatoy
on
Both sides.
dy) xlog)
x dlagx).
d
Diferen tiahom the uncton in the paramerifom
l e t x
=JCt)J=htt)a hoo diaeventiabk fanchons
a fsuch hotA eri>tk, then
(dy
dx

13. Sucecessive DYfeaetiahon
l e t y f(x) i dyfernfable funchon o x' its
dferen tiakle eoeLicient f ( x ) u also a
funcro in z'
- 8 's also digjerentiah
hen the denvatve fx)
wh eespec
to
c a l l e d 2" denahve o
w.r.t.
'x'. ha 2nd derivative of
w.T.t x' a
denoted
) )
dx
Similasly Y f dfernenhable
n' fimes hr.t 'z'
we deine
fCx)2) d'y (oe) x).jp
n" deevaftve Scme
Standard wncfions
s
fCa)
(axtb"/ mE z mlm m=n
m<n
2 (22+b)", mER mgn-1)Cm-a).
(m-ntI) Cax tby'm-n
nla" n=I,2, 3.4..n
3 ( a r t b ) + 1 .
Cn-)=1-2.3.. n-

14. No t()
gCax +b) Cn-)a"
( a z + b )
2!=Ix
2 2
! =
Ix2 x3=6
SSinCax+b) asinCaxtb+ ")H! /*2x3=h
6 CosCax+b) a cos(ax tbt
a a t b
C t e d
c"Coga). td
9. eSin Cbx t) + B ) e s i n ( b x t ctntan )
tol eos Cbx+e) tta4
C C b x
te t n tan b)
* Leibnitz's heorem
let fCx) E g[x) ane
two functiems
then
n-
aYe="fa)go) +"fgo)*
+CUVn
( u v ) , 6n u,V+nG
U-,V+..
. .
n
nC
-
n n
-1)

15. n! Ik2x3
n n-2!2!
n Cn-)
n C2
2
Formula
d(sin)
dx
casl) /1-z2
tan) = +
dr
Ccosec) =
dx
seclx) =
d
Ccot) / + 2

16. Pastial uleg
Lerentiation.
Partial Deatahves of odes
Let z=
4lz.y) be a funetion turo ndependen
vasiabes z
&y hen he parlal deartohve of 'z'
w.. derofed y difeseniatirg z'
Co.E by keepieg Y
a s c o r s t a n t .
s coel as parahal denivafives of2 cw.7.tH
is cdene fed E,2obtained difeentaty
wnty kp
a s Comstant
>Paztial Derivafives f 2 edea
lef 2-CLy) a fumehion oto
vaaiables &
a parfal desivatves o f& a e
ezst. Lf
ceas then hoy
he paath
derrvatives ef
d dzE
ase called 2 o2der pa~hal derivafvcA of 7:JCYy)
aherse a edenefesl

17. (D
yda
>Eule Theoren 02 honoqenous Funetionm
rehegenous funeÁons: A fCzH)
said fo be
homegenous fonetns degre D +
(Rx, ) = KACzy)
Eules 's heozen: Pu=fxy) a fmogeneus
tunchn degree n'i IH hen
Noe u=f(xy) sa homopenous functo deghee
n ry hen.
u--1)
au 0-1),
a)
+ T nn-i)u.
oy2
u
) 2

18. Mas common T2ignometricuneins uwed in the
the
ddfesntiaion
=>2sinz = l-Cas2
Cos 2c
=
/ - 2 S i n * x
=>
Sin2z = C e s 2
2
+Cos 2
Cos2z
= 2cos" -| = > 2cod
tcos2
2cod =
/+ts2
C o s =
2
cos 2A =cos2A -sthA
Sin(A +B) +Stn CA -B) = 2stmA osB
I - c o s z
=
2 Sin
I+Cos 2 Cos
tan
Sinx
COSX- Cot
8Secz =
I+cos2x
C o s 2 r =
Cos 2 2casr-1
Sin 89 =
3sino
-
hsin.
tan o SE O
secS- tan2 = /
Csec
Sin Costex
Sinz