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4 §1. ðÒÏÉÚ×ÏÄÎÁÑ ÆÕÎËÃÉÉ
òÅÛÅÎÉÅ. ëÁÖÄÁÑ ÉÚ ÄÁÎÎÙÈ ÆÕÎËÃÉÊ Ñ×ÌÑÅÔÓÑ ÓÔÅÐÅÎÎÏÊ ÆÕÎËÃÉÅÊ,
ÐÏÜÔÏÍÕ ×ÓÅ ÐÒÏÉÚ×ÏÄÎÙÅ ÎÁÈÏÄÑÔÓÑ ÐÏ ÆÏÒÍÕÌÅ (x
α
)
0
= αx
α−1
. éÍÅÅÍ:
x
17
0
= 17x
17−1
= 17x
16
;
x
π−e
0
= (π − e)x
π−e−1
;
1
x
0
=
x
−1
0
= −1 ·x
−1−1
= −x
−2
= −
1
x
2
;
√
x
0
=
x
1
2
0
=
1
2
x
1
2
−1
=
1
2
x
−
1
2
=
1
2
·
1
x
1
2
=
1
2
√
x
;
3
√
x
2
0
=
x
2
3
0
=
2
3
x
2
3
−1
=
2
3
x
−
1
3
=
2
3
·
1
x
1
3
=
2
3
3
√
x
;
1
5
√
x
7
0
=
1
x
7
5
0
=
x
−
7
5
0
= −
7
5
x
−
7
5
−1
= −
7
5
x
−
12
5
= −
7
5
·
1
x
12
5
= −
7
5
5
√
x
12
.
1.3. ðÒÏÉÚ×ÏÄÎÁÑ ÓÕÍÍÙ, ÒÁÚÎÏÓÔÉ, ÐÒÏÉÚ×ÅÄÅÎÉÑ É ÞÁÓÔÎÏÇÏ
ðÒÏÉÚ×ÏÄÎÙÅ ÓÕÍÍÙ, ÒÁÚÎÏÓÔÉ, ÐÒÏÉÚ×ÅÄÅÎÉÑ É ÞÁÓÔÎÏÇÏ Ä×ÕÈ ÆÕÎËÃÉÊ
u = u(x) É v = v(x) ÎÁÈÏÄÑÔÓÑ ÐÏ ÓÌÅÄÕÀÝÉÍ ÆÏÒÍÕÌÁÍ:
(u + v)
0
= u
0
+ v
0
, (u −v)
0
= u
0
− v
0
,
(uv)
0
= u
0
v + uv
0
, (cu)
0
= cu
0
, c ¡ ÞÉÓÌÏ,
u
v
0
=
u
0
v − uv
0
v
2
, v 6= 0.
ðÒÉÍÅÒ 5. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ
1
x
3
− 5 ln x.
òÅÛÅÎÉÅ.
1
x
3
− 5 ln x
0
=
1
x
3
0
− (5 ln x)
0
=
x
−3
0
− 5 (ln x)
0
=
= −3x
−4
− 5
1
x
= −
3
x
4
−
5
x
= −
3 + 5x
3
x
4
.
ðÒÉÍÅÒ 6. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ
2 ch x
3
+
cth x
4
.
òÅÛÅÎÉÅ.
2 ch x
3
+
cth x
4
0
=
2 ch x
3
0
+
cth x
4
0
=
2
3
(ch x)
0
+
1
4
(cth x)
0
=
=
2
3
sh x +
1
4
·
−
1
sh
2
x
=
2 sh x
3
−
1
4 sh
2
x
.
ðÒÉÍÅÒ 7. f (x) = 5x
3
− 3x
2
− 2x + 7, ÎÁÊÔÉ f
0
(0), f
0
(2), f
0
(−1).
4 §1. ðÒÏÉÚ×ÏÄÎÁÑ ÆÕÎËÃÉÉ
òÅÛÅÎÉÅ. ëÁÖÄÁÑ ÉÚ ÄÁÎÎÙÈ ÆÕÎËÃÉÊ Ñ×ÌÑÅÔÓÑ ÓÔÅÐÅÎÎÏÊ ÆÕÎËÃÉÅÊ,
ÐÏÜÔÏÍÕ ×ÓÅ ÐÒÏÉÚ×ÏÄÎÙÅ ÎÁÈÏÄÑÔÓÑ ÐÏ ÆÏÒÍÕÌÅ (xα )0 = αxα−1. éÍÅÅÍ:
0
x17 = 17x17−1 = 17x16;
0
xπ−e = (π − e)xπ−e−1;
0
1 0 1
= x−1 = −1 · x−1−1 = −x−2 = − 2 ;
x x
√ 0 1 0
1 1 1 1 1 1 1
x = x 2 = x 2 −1 = x− 2 = · 1 = √ ;
2 2 2 x2 2 x
√ 0 2 0 2 2 2 1 2 1 2
3
x2 = x 3 = x 3 −1 = x− 3 = · 1 = √ ;
3 3 3 x3 33x
0 0
1 1 − 57
0 7 − 7 −1 7 − 12 7 1 7
√ = 7 = x = − x 5 = − x 5 = − · = − √ .
5
x7 x5 5 5 5 x 125 5
5 x12
1.3. ðÒÏÉÚ×ÏÄÎÁÑ ÓÕÍÍÙ, ÒÁÚÎÏÓÔÉ, ÐÒÏÉÚ×ÅÄÅÎÉÑ É ÞÁÓÔÎÏÇÏ
ðÒÏÉÚ×ÏÄÎÙÅ ÓÕÍÍÙ, ÒÁÚÎÏÓÔÉ, ÐÒÏÉÚ×ÅÄÅÎÉÑ É ÞÁÓÔÎÏÇÏ Ä×ÕÈ ÆÕÎËÃÉÊ
u = u(x) É v = v(x) ÎÁÈÏÄÑÔÓÑ ÐÏ ÓÌÅÄÕÀÝÉÍ ÆÏÒÍÕÌÁÍ:
(u + v)0 = u0 + v 0, (u − v)0 = u0 − v 0 ,
(uv)0 = u0v + uv 0 , (cu)0 = cu0 , c ¡ ÞÉÓÌÏ,
u 0 u0v − uv 0
= , v 6= 0.
v v2
ðÒÉÍÅÒ 5. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ x13 − 5 ln x.
òÅÛÅÎÉÅ.
0 0
1 1 0 −3 0
− 5 (ln x)0 =
3
− 5 ln x = 3
− (5 ln x) = x
x x
1 3 5 3 + 5x3
= −3x−4 − 5 = − 4 − = − .
x x x x4
ðÒÉÍÅÒ 6. îÁÊÔÉ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ 2 ch3 x + cth4 x .
òÅÛÅÎÉÅ.
0 0 0
2 ch x cth x 2 ch x cth x 2 1
+ = + = (ch x)0 + (cth x)0 =
3 4 3 4 3 4
2 1 1 2 sh x 1
= sh x + · − 2 = − .
3 4 sh x 3 4 sh2 x
ðÒÉÍÅÒ 7. f (x) = 5x3 − 3x2 − 2x + 7, ÎÁÊÔÉ f 0(0), f 0(2), f 0 (−1).
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