text: 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.
explanation: The document appears to be a collection of calculus problems and solutions, specifically focusing on derivatives and integrals of various functions, including exponential, trigonometric, logarithmic, and inverse functions. Each problem is numbered and includes both the question and the corresponding solution.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: 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.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document appears to be a collection of calculus problems and solutions, specifically focusing on derivatives and integrals of various functions, including exponential, trigonometric, logarithmic, and inverse functions. Each problem is numbered and includes both the question and the corresponding solution.
  
  Label: multiple-question-and-answer