Производные А.С. Колпакова

tg x1

sin x x cos x

aa

xa

ax

1

1

xp3

p2

1

x

a

2x

1+x

x2

(2 x2)(3 x3)

1=2

1=3

1+x3

3

2

2

a

x

b

a

x

b

1:

2

;

2:

;

3: (1+x)(2+x)

(3+x)

;

4:

3

; 5: sin (cos

x) cos

sin

x ; 6:

2

;

2

;

1 x+x

8

(1 x)

;

4

+

ln

1 x

11 ln tg

cos

ln

x

a

b

7: 2

;

:

9: x +a +a

;

10:

:

x

tg x;

12: arcctg

;

4(1+x ) 4

2

cos x+x sin x

xp3+p2

2pax x

p

2

2

q

p

2

ln x

2

13: p

x + 1

ln(1 + p

x + 1);

14:

cos x

+ ln

1+cos x

;

15:

x ln(x + px2

+ a2);

16: arcsin (sin x

cos x) ;

2

sin x

2 sin

x

2

17: x+

1 x

arccos x; 18: ln(1+sin

x) 2 sin x arctg(sin x); 19: arctg

x

1

p

; 20: arctg x ln (1 + x ) ;

x2

1

arcsin(sin2 x)

arctg2 x

2

3

(x3+1)(x 7)2

cossin x x

2

(sin x+1)(cos x 10)

tg(ex

)

21:

;

22: ln(ln

(ln

x));

23:

;

24:

sincos x x

;

25:

;

26:

;

arccos(cos2 x)

(x2+5)3

cos4(sin(ln x))

(tg

2

x+5)(e

x2

1)

q

2

p

2

p

2

p

2

pa +bx

b+a cos x+pb2

a2 sin x

(ln x)x

32

p

1 +

) 2

1 +

ln(p+

29: eax

pa2 x2

;

a+b cos x

;

31:

2

;

xln x

27:

(1

x)m(1 + x)n;

28:

x +

x + px;

a sin bx

b cos bx

30: x(sin(ln x)

cos(ln x));

sin

x

;

m+n

2

2

sin(x2)

: x ln (x +

x

x

x

1 + x ) + 2x; 33:

; 34: ln

q1

35:

1+

:

arctg

a b

x

;

40:

arcctg x6;

:

x

42:

ln

4

1+x +x

;

p

p

p

1

(x+1)2

1

2x 1

b

2p

a2

b2

a b

x

x

x

;

39

pa2

b2

a+b tg

2

1+x12

;

4

p1+x4

x

2 arctg

x

41

arctg 1+p1 x2

36: x arcsin

1+x + arctg

x;

37:

6 ln x2 x+1 + p3 arctg

p3

;

38:

a x +

a

arctg

a+b th

2

q

tg3 x

x6

tg5 x

2

p

2

4

1

4

2

1

4

x4

p

p

43: p

arctg p

;

44: tg x

3 +

; 45: x(arcsin x)

+ 2 1 x

arcsin x 2x;

46:

p

(x+p

)

;

x

x

5

1+x2

1+x2

47: x2 p

+

a2

arcsin xa :