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# Turunan Fungsi Eksponen

x
= e sin 2 x 3

1. y

dy

2. y

dy

dx

e x sin 2 x 3 d

dx

sin 2 x 3.e x

x
= e sin 2 x 3 2 cos 2 x 3

= e 5 x 2 2 x 1

= e 5 x 2 x 1 dy
5x 2 2 x 1
dx
dx
2
= (10x + 2) e 5 x 2 x 1
2

## Turunan Fungsi Trigonometri

= sin (2x2+3) + cos 3x
3. y
dy/dx = cos (2x2+3) d/dx (2x2+3) + (sin 3x) d/dx (3x)
= 4x cos (2x2+3) 3sin 3x
2
4. Y = sec 5x
misalkan u = sec 5x Y = u2
dy/du = 2 u ; v = 5x
du/dx = sec v tg v dv/dx = 5
dy/dx = dy/du . du/dv . dv/dx
= 10 sec25x tg 5x

5. y

dy

= log 5 sin 2x

## dx = log 5e . 2 sin x. cos x

sin 2x
= log 5e . 2 cos x
sin x
= 2 ctg x log 5e

= In 3x3 + In 2 x
1
1
1
dy
9x 2
.
3
2x
x
dx = 3x
= 4/x

6. y

7. y
dy
dx

x
= 2log 5

1
= 5

x2 2

d x2 2
5
In 2 dx

5 . In 5. 2x
In 2

5x

x2 2

= 2x In 5
In 2
Invers Fungsi
8. Diketahui f ( x) x 1
x2
a. Periksa apakah f mempunyai invers
Jawab :
a.

3
1.( x 2) 1.( x 1)

0, x Df
2
( x 2) 2
(
x

2
)
Karena f selalu naik(monoton murni) maka f mempunyai invers

f ' ( x)

b. Misal, y

x 1
x 2

2y 1
xy 2 y
xx
xy
12 y 1 x
y 1
f

( y)

## Turunan Fungsi Invers

9. Diketahui,
Jawab :

2y 1
y 1

f ( x ) x 5 2 x 1 , tentukan : ( f

( x)

2x 1
x 1

)' (4)

## ,y=4 jika hanya jika x=1

f ' ( x) 5 x 4 2

1
1
( f 1 )' (4)

f ' (1) 7
Fungsi Logaritma Asli

## 10. Diberikan f ( x ) ln(sin(4 x 2))

Maka
Jika

f ' ( x)

1
Dx (sin(4 x 2)) 4 cos(4 x 2)
sin( 4 x 2)

y ln | x | , x 0

ln x , x 0

ln( x ) , x 0

y ln x y '

y ln( x) y '

d
1
(ln | x |)
, x 0.
dx
x
Dari Sini Diperoleh :
1
dx ln | x | C

x
11. 4 x 2
0 x 3 2dx
u x 3 2 du 3 x 2 dx

x2
x 2 du 1 1
1
dx

x3 2
u 3x 2 3 u du 3 ln | u | c
4
4
x2
1
3
1
1
0 x 3 2dx 31 ln
ln || xx 3 22 || 0c (ln 66 ln 2) ln 33.
3
3
3

1
x

1
1

x
x

2 x 1
2sin 2 x
12. f ( x ) 3

## f ' ( x ) 2.32 x 1 ln 3 2.2sin 2 x cos 2 x ln 2

13.

x2

.xdx

u x 2 du 2 xdx dx 21x du
2

du 1 4 u
4x
4 .xdx 4 2 2 ln 4 C 2 ln 4 C
Fungsi Logaritma Umum
x2

3
2
14. f ( x ) log( x 1)

f ' ( x)
15.

ln( x 2 1)
ln 3

2x
1
x 1 ln 3
2

ln( xx 11 )
x 1
)
x 1
ln 4
1 1
x 1
f ' ( x)
Dx
(
)
ln 4 xx 11
x 1
f ( x) 4 log(

1 x 1 x 1 ( x 1)
ln 4 x 1 ( x 1) 2

1
2
ln 4 ( x 1)( x 1)

### Dapatkan aplikasi gratis kami

Hak cipta © 2021 Scribd Inc.
The text passage contains a series of mathematical problems and their solutions, primarily focusing on derivatives and integrals of various functions, including exponential, trigonometric, logarithmic, and inverse functions.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Anda di halaman 1dari 3

# Turunan Fungsi Eksponen

x
= e sin 2 x 3

1. y

dy

2. y

dy

dx

e x sin 2 x 3 d

dx

sin 2 x 3.e x

x
= e sin 2 x 3 2 cos 2 x 3

= e 5 x 2 2 x 1

= e 5 x 2 x 1 dy
5x 2 2 x 1
dx
dx
2
= (10x + 2) e 5 x 2 x 1
2

## Turunan Fungsi Trigonometri

= sin (2x2+3) + cos 3x
3. y
dy/dx = cos (2x2+3) d/dx (2x2+3) + (sin 3x) d/dx (3x)
= 4x cos (2x2+3) 3sin 3x
2
4. Y = sec 5x
misalkan u = sec 5x Y = u2
dy/du = 2 u ; v = 5x
du/dx = sec v tg v dv/dx = 5
dy/dx = dy/du . du/dv . dv/dx
= 10 sec25x tg 5x

5. y

dy

= log 5 sin 2x

## dx = log 5e . 2 sin x. cos x

sin 2x
= log 5e . 2 cos x
sin x
= 2 ctg x log 5e

= In 3x3 + In 2 x
1
1
1
dy
9x 2
.
3
2x
x
dx = 3x
= 4/x

6. y

7. y
dy
dx

x
= 2log 5

1
= 5

x2 2

d x2 2
5
In 2 dx

5 . In 5. 2x
In 2

5x

x2 2

= 2x In 5
In 2
Invers Fungsi
8. Diketahui f ( x) x 1
x2
a. Periksa apakah f mempunyai invers
Jawab :
a.

3
1.( x 2) 1.( x 1)

0, x Df
2
( x 2) 2
(
x

2
)
Karena f selalu naik(monoton murni) maka f mempunyai invers

f ' ( x)

b. Misal, y

x 1
x 2

2y 1
xy 2 y
xx
xy
12 y 1 x
y 1
f

( y)

## Turunan Fungsi Invers

9. Diketahui,
Jawab :

2y 1
y 1

f ( x ) x 5 2 x 1 , tentukan : ( f

( x)

2x 1
x 1

)' (4)

## ,y=4 jika hanya jika x=1

f ' ( x) 5 x 4 2

1
1
( f 1 )' (4)

f ' (1) 7
Fungsi Logaritma Asli

## 10. Diberikan f ( x ) ln(sin(4 x 2))

Maka
Jika

f ' ( x)

1
Dx (sin(4 x 2)) 4 cos(4 x 2)
sin( 4 x 2)

y ln | x | , x 0

ln x , x 0

ln( x ) , x 0

y ln x y '

y ln( x) y '

d
1
(ln | x |)
, x 0.
dx
x
Dari Sini Diperoleh :
1
dx ln | x | C

x
11. 4 x 2
0 x 3 2dx
u x 3 2 du 3 x 2 dx

x2
x 2 du 1 1
1
dx

x3 2
u 3x 2 3 u du 3 ln | u | c
4
4
x2
1
3
1
1
0 x 3 2dx 31 ln
ln || xx 3 22 || 0c (ln 66 ln 2) ln 33.
3
3
3

1
x

1
1

x
x

2 x 1
2sin 2 x
12. f ( x ) 3

## f ' ( x ) 2.32 x 1 ln 3 2.2sin 2 x cos 2 x ln 2

13.

x2

.xdx

u x 2 du 2 xdx dx 21x du
2

du 1 4 u
4x
4 .xdx 4 2 2 ln 4 C 2 ln 4 C
Fungsi Logaritma Umum
x2

3
2
14. f ( x ) log( x 1)

f ' ( x)
15.

ln( x 2 1)
ln 3

2x
1
x 1 ln 3
2

ln( xx 11 )
x 1
)
x 1
ln 4
1 1
x 1
f ' ( x)
Dx
(
)
ln 4 xx 11
x 1
f ( x) 4 log(

1 x 1 x 1 ( x 1)
ln 4 x 1 ( x 1) 2

1
2
ln 4 ( x 1)( x 1)

### Dapatkan aplikasi gratis kami

Hak cipta © 2021 Scribd Inc.
user
The text passage contains a series of mathematical problems and their solutions, primarily focusing on derivatives and integrals of various functions, including exponential, trigonometric, logarithmic, and inverse functions.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS