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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)
  
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