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2 §1. ðÒÏÉÚ×ÏÄÎÁÑ ÆÕÎËÃÉÉ
ÔÏ y(x
0
) =
1
x
0
É y(x
0
+ 4x) =
1
x
0
+4x
, ÐÏÜÔÏÍÕ
4 y(x
0
) = y(x
0
+ 4x) − y(x
0
) =
1
x
0
+ 4x
−
1
x
0
=
=
x
0
− (x
0
+ 4x)
x
0
(x
0
+ 4x)
=
− 4 x
x
0
(x
0
+ 4x)
.
ïÔÓÀÄÁ,
4y(x
0
)
4x
=
−4x
x
0
(x
0
+4x)
4x
= −
1
x
0
(x
0
+ 4x)
.
óÌÅÄÏ×ÁÔÅÌØÎÏ,
y
0
(x
0
) = lim
4x→0
4y(x
0
)
4x
= lim
4x→0
−
1
x
0
(x
0
+ 4x)
= − lim
4x→0
1
x
2
0
+ x
0
4 x
= −
1
x
2
0
.
ôÁË ËÁË × ËÁÞÅÓÔ×Å x
0
ÍÏÖÎÏ ×ÚÑÔØ ÌÀÂÏÅ ÞÉÓÌÏ ÎÅÒÁ×ÎÏÅ ÎÕÌÀ, ÔÏ ÄÌÑ
ÌÀÂÏÇÏ ÞÉÓÌÁ x 6= 0 ÐÏÌÕÞÁÅÍ
y
0
(x) =
1
x
0
= −
1
x
2
.
îÁÐÒÉÍÅÒ, y
0
(−2) = −
1
(−2)
2
= −
1
4
.
ðÒÉÍÅÒ 2. îÁÊÔÉ ÐÏ ÏÐÒÅÄÅÌÅÎÉÀ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ y = sin x.
òÅÛÅÎÉÅ. úÁÆÉËÓÉÒÕÅÍ ÐÒÏÉÚ×ÏÌØÎÕÀ ÔÏÞËÕ x
0
. ôÁË ËÁË y(x) = sin x,
ÔÏ y(x
0
) = sin x
0
É y(x
0
+ 4x) = sin(x
0
+ 4x), ÐÏÜÔÏÍÕ
4y(x
0
) = y(x
0
+ 4x) − y(x
0
) = sin(x
0
+ 4x) − sin x
0
.
óÌÅÄÏ×ÁÔÅÌØÎÏ,
4y(x
0
)
4x
=
sin(x
0
+ 4x) − sin x
0
4x
=
=
2 sin
4x
2
cos
x
0
+
4x
2
4x
=
sin
4x
2
4x
2
cos
x
0
+
4x
2
.
÷ÏÓÐÏÌØÚÏ×Á×ÛÉÓØ ÎÅÐÒÅÒÙ×ÎÏÓÔØÀ ÆÕÎËÃÉÉ sin x É ÐÅÒ×ÙÍ ÚÁÍÅÞÁÔÅÌØÎÙÍ
ÐÒÅÄÅÌÏÍ
lim
4x→0
sin
4x
2
4x
2
= lim
α→0
sin α
α
= 1,
ÐÏÌÕÞÁÅÍ
y
0
(x
0
) = lim
4x→0
4y(x
0
)
4x
= 1 · cos x
0
= cos x
0
.
2 §1. ðÒÏÉÚ×ÏÄÎÁÑ ÆÕÎËÃÉÉ
1 1
ÔÏ y(x0 ) = x0
É y(x0 + 4x) = x0 +4x
, ÐÏÜÔÏÍÕ
1 1
4 y(x0) = y(x0 + 4x) − y(x0) = − =
x0 + 4x x0
x0 − (x0 + 4x) −4x
= = .
x0(x0 + 4x) x0 (x0 + 4x)
ïÔÓÀÄÁ,
−4x
4y(x0 ) x0 (x0 +4x) 1
= =− .
4x 4x x0(x0 + 4x)
óÌÅÄÏ×ÁÔÅÌØÎÏ,
0 4y(x0) 1 1 1
y (x0) = lim = lim − = − lim 2 = − 2.
4x→0 4x 4x→0 x0(x0 + 4x) 4x→0 x0 + x0 4 x x0
ôÁË ËÁË × ËÁÞÅÓÔ×Å x0 ÍÏÖÎÏ ×ÚÑÔØ ÌÀÂÏÅ ÞÉÓÌÏ ÎÅÒÁ×ÎÏÅ ÎÕÌÀ, ÔÏ ÄÌÑ
ÌÀÂÏÇÏ ÞÉÓÌÁ x 6= 0 ÐÏÌÕÞÁÅÍ
0
0 1 1
y (x) = = − 2.
x x
1 1
îÁÐÒÉÍÅÒ, y 0 (−2) = − (−2) 2 = −4.
ðÒÉÍÅÒ 2. îÁÊÔÉ ÐÏ ÏÐÒÅÄÅÌÅÎÉÀ ÐÒÏÉÚ×ÏÄÎÕÀ ÆÕÎËÃÉÉ y = sin x.
òÅÛÅÎÉÅ. úÁÆÉËÓÉÒÕÅÍ ÐÒÏÉÚ×ÏÌØÎÕÀ ÔÏÞËÕ x0. ôÁË ËÁË y(x) = sin x,
ÔÏ y(x0 ) = sin x0 É y(x0 + 4x) = sin(x0 + 4x), ÐÏÜÔÏÍÕ
4y(x0) = y(x0 + 4x) − y(x0) = sin(x0 + 4x) − sin x0 .
óÌÅÄÏ×ÁÔÅÌØÎÏ,
4y(x0 ) sin(x0 + 4x) − sin x0
= =
4x 4x
4x 4x
2 sin 2 cos x0 + 2 sin 4x
4x
2
= = 4x
cos x0 + .
4x 2
2
÷ÏÓÐÏÌØÚÏ×Á×ÛÉÓØ ÎÅÐÒÅÒÙ×ÎÏÓÔØÀ ÆÕÎËÃÉÉ sin x É ÐÅÒ×ÙÍ ÚÁÍÅÞÁÔÅÌØÎÙÍ
ÐÒÅÄÅÌÏÍ
sin 4x sin α
lim 4x2 = lim = 1,
4x→0 α→0 α
2
ÐÏÌÕÞÁÅÍ
4y(x0)
y 0 (x0) = lim = 1 · cos x0 = cos x0.
4x→0 4x
