R xyc/kpi/ma/2020-2021/i//ka-03...cos2 x shx chx e xe ctgx − 1 sin2 x chx shx log a|x| 1 xln...

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�� X ⊂ R� f : X → R � x0 ∈ R � �� X� �� f �� x0 ��

�� x0 ��

�� f �

�� X ⊂ R� x0 ∈ R � �� f : X → R� �� x0 ��

�� X ∩ (−∞, x0) �= ∅� X ∩ (x0,+∞) �= ∅ � �� f(x0 − 0)� f(x0 + 0)�

�� X ∩ (−∞, x0) �= ∅� X ∩ (x0,+∞) = ∅� x0 ∈ X� � �� f(x0 − 0)�

�� X ∩ (−∞, x0) = ∅� X ∩ (x0,+∞) �= ∅� x0 ∈ X� � �� f(x0 + 0)�

�� x0 � �� f(x0 +0)−f(x0 − 0) �� f(x0)− f(x0 − 0) �� f(x0+0)−f(x0) �� f � �� x0�

�� f(x0−0) = f(x0+0) �� x0 ��

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y =x2 − 1

x3 − 3x+ 2 .

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y =x

sinx.

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y = sgn�sin

π

x

�.

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�� f : X → R x0 ∈ X � �� X� ��

f �(x0) = limx→x0

f(x)− f(x0)x− x0

= limΔx→0

f(x0 +Δx)− f(x0)Δx

,

�� f � ��

x0�

x

y

�

x0

f(x0)

x0 + Δx

f(x0 + Δx) �

Δx

Δy

αM0

M

(a)

x

y

�

x0

f(x0)α0M0

(b)

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�� f, g �� x0� � C ∈ R� �� f + g� Cf � fg�� x0� �� g(x0) �= 0� �� fg ��

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�� x0� ��

(f + g)�(x0) = f�(x0) + g

�(x0), ��

(Cf)�(x0) = Cf�(x0), ��

(fg)�(x0) = f�(x0)g(x0) + f(x0)g

�(x0), ��fg

��(x0) =

f �(x0)g(x0)− f(x0)g�(x0)g2(x0)

. ��

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f(x) f �(x) f(x) f �(x) f(x) f �(x)

C 0 sinx cosx arctg x 11+x2

xα αxα−1 cosx − sinx arcctg x − 11+x2

ax ax ln a tg x 1cos2 x

shx chx

ex ex ctg x − 1sin2 x

chx shx

loga |x| 1x ln a arcsinx 1√1−x2 thx1

ch2 x

ln |x| 1x

arccosx − 1√1−x2

cthx − 1sh2 x

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�� f : X → Y � g : Y → R�x0 ∈ X � �� X� � y0 = f(x0) � �� Y � �� f �� x0� � g �� y0� �� g ◦ f : X → R �� x0� �

(g ◦ f)�(x0) = g�(y0)f �(x0).

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�� y�(0)� y�( 12 )� y�(−10)� ��

y = 2 + x− x2.

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y = a5 + 5a3x2 − x5.

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y =1

x+

2

x2+

3

x3.

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y = x+√x+ 3

√x.

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y =2x

1− x2 .

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y =3

�1 + x2

1− x2 .

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y =

�x+

�x+

√x.

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y = cos 2x− 2 sinx.

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y = sin(cos2 x) cos(sin2 x).

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y = e−x2

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y = x+ xx + xxx

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y = x√x.

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y =x2

1− x3

�3− x

(3 + x)2.

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y = (x− a1)n1(x− a2)n2 · . . . · (x− ak)nk .

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f(x) =

�x2 sin 1x , x �= 0,0, x = 0

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R xyc/kpi/ma/2020-2021/i//ka-03...cos2 x shx chx e xe ctgx − 1 sin2 x chx shx log a|x| 1 xln...

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