Математика / 1 семестр / Модуль 6. Производная / Производная 1

u v = u

v ,

u v = u

v u v ,

C u = C u .

1

3

3

3

5/3

2

5

= arcsin x x

2x 3x

x

(x

2)(x

) =

7

1 x2

x

6

x 3x2 3

(x3 2)

5

(x2/3) =

= arcsin x

x5

x2

1

7

3

= arcsin x

x

6

x 3x11/3

5

(x3 2)x2/3.

x2

7

3

1

1

1/8

2

3x

3.

y =

x

arctgx

.

x3

6

x

1

1/8

1/8

1

(3

x

3

x

(

) x

x)

y

=

(x

)

arctg x x

(arctg x)

2

=

6

x3

(

x

)2

3x ln 3

3x

x

1

1

7/8 arctgx x1/8

1

x 3

2

x

=

x

2

=

2

6

1x

x

8

1

arctg x

x1/8

2

4

3x ln 3 2x 3x

3x

=

=

6

8x

7/8

1 x

2

2x x

1

arctg x

x1/8

6 3x (2x ln 3 1)

=

7/8

2

4

3/2

.

6

8x

1 x

x

2x

y

=

(a

u

u

= 10

x tg 3x

ln10

x tg 3x =

) = a

ln a u ,

u

(3x)

=

tg u =

= 10x tg 3x ln10 tg 3x x

=

cos2 u

cos2 3x

1

= 10x tg 3x ln10 tg 3x x

3 .

cos2 3x

5. y = ctg

lg 3

1

sin2 5x

.

4

arcsin

1

x

u

u v v u

y

=

=

=

(C) = 0, (u v) = u

v ,

v2

v

1 (sin

2

arcsin

1

sin

2

5x (arcsin

1

= 0

5x)

x

x

)

=

4

arcsin

2 1

2

x

1

=

(u

) = 2u

u , (arcsin u) =

1 u2

u

=

1

2

1

1

2sin 5x (sin 5x) arcsin

sin

5x

1 (1/x)2

=

1

x

x

=

2 1

4

arcsin

x

1

2

1

1

2sin 5x cos5x 5 arcsin

sin

5x

2

x

2

=

1

1 (1/x)

x

.

4

2 1

arcsin

x