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Hosilaning geometrik ma’nosi

^{y  f}  ^x
funksiya grafigiga
x  x₀
nuqtada o‘tkazilgan urinmaning burchak

koeffitsienti
k  tgα  y^ (x₀ ) .

Urinma tenglamasi:


y  f (x₀ ) 
^{f '(x0)} ^{x  x0} ^.

Hosilaning fizik ma’nosi

Harakatlanayotgan jismning t₀
vaqtda bosib o‘tgan yo‘li
S  f (t ) , uning t₀

vaqtdagi tezligi V (t₀ ) , tezlanishi esa
V (t₀ ) 
a(t₀ ) f ^(t₀ ) ,
bo‘lsa, u holda
a(t₀ )  V ^(t₀ ) .

Differensiallashning asosiy qoidalari

^{u  v} ^{  u  v ;} ^{u  v}^{  uv  uv;}

u ^ u^v  uv^{ }

 _v 
  _{ ;}
  ^v2
 ^f ^{g  x}
 f ^_ g  x_ g^ x .

Funksiyani hosila yordamida tekshirish

^{y  f}  ^x
funksiya (a,b)
oraliqda aniqlangan bo‘lsin.

1. 
   Agar (a,b)
   
2. 
   Agar (a,b)
   

oraliqda oraliqda
y^  0
y^  0
bo‘lsa, u holda funksiya shu oraliqda o‘suvchi; bo‘lsa, u holda funksiya shu oraliqda kamayuvchi;

3. 
   Agar (a,b)
   

oraliqda
y^  0
bo‘lsa, u holda funksiya shu oraliqda botiq, agar
y^  0

bo‘lsa, shu oraliqda qavariq bo‘ladi.

^{f '} ^x ^{ 0}

Funksiyaning eng katta va eng kichik qiymatlari

tenglamaning ildizlari va hosila mavjud bo‘lmagan nuqtalar

^{y  f}  ^x

funksiyaning statsionar nuqtalari deyiladi.
^a,b ^{kesmada uzluksiz bo‘lgan y  f}  ^x ^{funksiyaning shu kesmadagi eng katta}

^{va eng kichik qiymatlari f}  ^x1 ^{, f}  ^x2 ^{, . . ., f}  ^xn ^{, f} ^a^,
f ^b ^{sonlar orasida}

^{bo‘ladi. Bu yerda x1, x2 ,... xn lar y  f}  ^x ^funksiyaning ^a,b
nuqtalari.
oraliqdagi statsionar

Elementar funksiyalarning hosilalari

Funksiya	Hosilasi
y  x	y^ 1
y  ctgkx	y^  ^k Sin^2kx
y  arcsin kx	y^^k ^1^^kx^2
y  arccos kx	y^  ^k 1  (kx)^2
y  arctgkx	y^ ^k 1  (kx)^2
y  arcctgkx	y^  ^k 1  (kx)^2
y  e^x	y^ e^x
y  ag ( x )	y^ a^g^(^x^) ln a  g^(x)
y  tgf (x)	y  g^(x) Cos^2 f (x)
y  ctgg(x)	y   g^(x) Sin^2g(x)
y  arcS in f (x)	y  f '(x) 1  f ^2 (x)
y  arcCosf (x)	y   f '(x) 1  f ^2 (x)
y  arctgf (x)	y  f '(x) 1  f ^2(x)
y  arcctgg(x)	y   g^(x) 1  g ^2(x)

Funksiya	Hosilasi
y  С	y^ 0
y  x^m	y^ m  x^m^^1
y  a^x	y^ a^x  ln a
y  loga x	y^ ^1 x ln a
y  Sinkx	y^ k  Coskx
y  Coskx	y^ k  Sinkx
y  tgkx	y^ ^k Cos^2kx
^y^^g^(^x^)^n	^y^^^n^g^(^x^)^n^^1^g^^(^x^)
y  ^1 g^n (x)	y   ng^(x) g^n^^1(x)
y  ^ng(x)	y  g^(x) n^n g^n^^1(x)
y  1 ^ng(x)	y   g^(x) n^n g^n^^1(x)
y  Sing(x)	y^ Cosg(x)  g^(x)
y  Cosf (x)	y^ Sinf (x)  f '(x)
^y^loga^g^(^x^)	y  g^(x) g(x) ln a

Ajoyib limitlar

1. lim ^Sinx  lim ^x
 1. 2.
lim x^x  1.

x0 _x x0 _Sinx
Sinpx
x0
1

3. lim
x0 _x
 p,
   p   . 4.
lim(1  x) ^x x0
 e .

5. lim
x0
C^x 1



x
 ln C,
C  0 . 6.


lim
x0
tgx x
 1.


7. lim ^1 
n _{
1 }n

n


 e  2, 7183....

Boshlang‘ich funksiya (integral) Boshlang‘ich funksiya va uni topishning sodda qoidalari

Agar berilgan oraliqdagi barcha x uchun
F ^(x) 
^f  ^x
tenglik bajarilsa, u holda

F (x)
funksiya shu oraliqda
f (x)
funksiyaning boshlang‘ich funksiyasi deyiladi.

Agar
F (x)
funksiya
y  f (x)
funksiyaning boshlang‘ich funksiyasi bo‘lsa:

1. 
   har qanday o‘zgarmas S uchun bo‘ladi;
   

F (x)  С
ham
f (x)
ning boshlang‘ich funksiyasi

2. 
   ¹ F (kx  b) k
   

funksiya
y  f (kx  b)
funksiyaning boshlang‘ich funksiyasi bo‘ladi.

Elementar funksiyalarning boshlang‘ichlari

Funksiya		Boshlang‘ichi				Funksiya	Boshlang‘ichi
y  x^m m  1		m1 Y  ^x m  1				y  ctgkx	Y  ^1ln Sinkx  C k
y  a^x		Y	 ^ax  ln a	C		y  ^1 Sinx	Y  ln tg ^x C 2
y  ^1 x		Y  ln x  C				y  ^1 Cosx	Y  ln tg ^^x ^π^ C  2 2   
y  Sinkx		Y   ^1 Coskx  C k				y  ^1 Sin^2x	Y  ctgx  C
y  Coskx		Y  ^1 Sinkx  C k				y  ^1 Cos^2x	Y  tgx  C
y  tgkx		Y   ^1ln Coskx  C k				y  ^1 a^2  x^2	Y  ^1arctg ^x C a a
y 	1 x^2  a^2	Y  ^1ln ^x^^a C 2a x  a				y  ^1 a^2  x^2	Y  arcsin ^x C a
y  e^x		Y  e^x  С				y  1 x^2  a^2	Y  ln x 	x^2  a^2  C