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IR
fPFpI, Rqf a PI f1paq
lim
xÑa
fpxq ´ fpaq
x´a“f1paq p1q
fpxq “ fpaq`px´aqf1paq ` oppx´aqq p2q
f1paq pa, fpaqq
px, fpxqq x a
Cf
xa
fpaq
fpxq
T
dptqt
dptq ´ dpt0q
t´t0
loooooomoooooon
pt t0q
ÝÑ
tÑt0
vpt0q “ d1pt0q
looooooomooooooon
pt0q
h“x´ap1q
lim
hÑ0
fpa`hq ´ fpaq
h“f1paq p1q
x“a`hp2q
fpa`hq “ fpaq ` hf1paq ` ophq p2q
fPFpI, Cqf a
g“Re f h “Im f a f1paq “ g1paq ` ih1paq
f a f pTqM0pa, fpaqq
y“fpaq`px´aqf1paq
a
xpCfq pTq
"fpxq “ fpaq`px´aqf1paq ` oppx´aqq
y“fpaq`px´aqf1paq

fpxq ´ y“opx´aq
x a
fPFpI, Rqa f a
f a
fpxq “ fpaq`px´aqf1paq ` oppx´aqq ÝÑ
xÑafpaq
f a
f:"RÑR
xÞÑ |x|0 0
‚xą0fpxq ´ fp0q
x´0“|x| x“1ÝÑ
xÑ01
‚xă0fpxq ´ fp0q
x´0“|x| x“ ´1ÝÑ
xÑ0´1‰1
fPFpI, Rqf a PI
fpxq ´ fpaq
x´a
a f1
dpaqf1 gpaq
a0
f a
pa, fpaqq
Cf
a
fpaq
f a
a f1
dpaq “ f1 gpaq
f:xÑ#xsin 1
xx‰0
0x“0
0x‰0
fpxq ´ fp0q
x´0“sin 1 x
sin 1
xx0
fPFpI, Rqf I f|I
I x ÞÑ f1pxqf f1, Df df
dx

@xPR, fpxq “ C f x0PR
fpxq ´ fpx0q
x´x0“C´C x´x0“0ÝÑ
xÑx0
0
@xPR, f1pxq “ 0
@xPR, fpxq “ x x0PR
fpxq ´ fpx0q
x´x0“x´x0 x´x0“1ÝÑ
xÑx0
1
@xPR, f1pxq “ 1
@xPR`, fpxq “ ?x x0PR˚
`
fpxq ´ fpx0q
x´x0“?x´?x0 x´x0“?x´?x0 p?x´?x0qp?x`?x0q“1
?x`?x0ÝÑ
xÑx0
1
2?x0
@xPR˚
`, f1pxq “ 1
2?x
f g x0λPR
f`g x0pf`gq1px0q “ f1px0q ` g1px0q
λf x0pλfq1px0q “ λf1px0q
fg x0pfgq1px0q “ f1px0qgpx0q ` fpx0qg1px0q
fpx0q ‰ 01
fx0ˆ1 f˙1
px0q “ ´f1px0q
f2px0q
gpx0q ‰ 0f
gx0ˆf g˙1
px0q “ f1px0qgpx0q ´ fpx0qg1px0q
g2px0q
1
fpxq´1 fpx0q
x´x0
“1
fpxqfpx0qˆfpx0q ´ fpxq x´x0
ÝÑ
xÑx0
1
f2px0qˆ p´f1px0qq
f a g fpaq
g˝f a
pg˝fq1paq “ g1pfpaqq ˆ f1paq
f a
fpxq “ fpaq`px´aqf1paq ` opx´aq p1q
g fpaq
gpyq “ gpfpaqq ` py´fpaqqg1pfpaqq ` opy´fpaqq p2q
y“fpxq p2q
pg˝fqpxq “ gpfpxqq
“gpfpaqq ` `fpxq ´ fpaq˘g1pfpaqq ` opfpxq ´ fpaqq
“gpfpaqq ` `px´aqf1paq ` 0px´aq˘g1pfpaqq ` o`px´aqf1paq ` 0px´aq˘
pg˝fqpxq “ gpfpaqq ` px´aqf1paqg1pfpaqq ` opx´aq
lim
xÑa
pg˝fqpxq´pg˝fqpaq
x´a“f1paqg1pfpaqq
f I a PI
f1paq ‰ 0f´1b“fpaq
pf´1q1pbq “ 1
f1`f´1pbq˘

yPfpIqx“f´1pyq ô y“fpxqyÑbôxÑa
f´1pyq ´ f´1pbq
y´b“x´a fpxq ´ fpaqÝÑ
yÑb
1
f1paq“1 f1pf´1pbqq
f I
"fp0q“f
fpkq“`fpk´1q˘1fpk´1q
fp1q“f1fp2q“f2fpnqn f
Dkfdkf
dxk
fCnI f n I
fC0f f C1f
f1CnpIqCnI
f k f, f1, . . . , fpk´1qfpkq
f:#xÞÑ x2sin 1
xx‰0
0ÞÑ 0
Rf10f
R˚
fpxq ´ fp0q
x´0“xsin 1 xÝÑ
xÑ00
f0f1p0q “ 0x‰0
f1pxq “ 2xsin 1
x´cos 1 x
0 2xsin 1
xÝÑ
xÑ00 cos 1
x0f1
0
fC8fCnnPN
C8IC8pIq
fC8I f
f g n CnλPR
f`g, λf f g n Cn
pf`gqpnq“fpnq`gpnq
pλfqpnq“λfpnq
pfgqpnq“
n
ÿ
k“0ˆn
k˙fpkqgpn´kq
CkpIq,C8pIqCpIq

n
f:xÞÑ p3x2`x´5qe´x
g:xÞÑ x3sin x
h:xÞÑ cos3xsin2x
f:IÑJ g :JÑRCng˝f:IÑRCn
n f g Cn
˝n“0
˝nCnCnf
gCn`1f1g1Cng˝f
pg˝fq1“f1ˆ pg1˝fq
g1fCng1˝fCnpg˝fq1Cng˝f
Cn`1
˝ně0
fra, bsa‰bsa, br
fpaq “ fpbqcPsa, brf1pcq “ 0
Cf
b
a c
fpaq “ fpbq
RÑC
f:xÞÑ eixfp0q “ fp2πq “ 1
@xPs0,2πr, f1pxq “ ieix‰0
PPRrXsP P 1
f
ra, bsa‰bsa, brcPsa, br
fpbq ´ fpaq“pb´aqf1pcq
Cf
b
a c
fpaq
fpbq
f I
Ipα, βq P I c Psα, βr
fpβq ´ fpαq“pβ´αqf1pcq