comp diff

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comp diff

  1. 1. Composite functions
  2. 2. f(x) = x 2 x x 2 2 4 7 49 g(x) = x + 1 2 3
  3. 3. f(g(x)) f(x) = x 2 g(x) (g(x)) 2 x g(x) = x + 1 x x+1 (x+1) 2
  4. 5. f x x f x ( ) ( ) ( ) ? = + Þ ¢ = 1 2 f x x x x ( ) ( ) = + = + + 1 2 1 2 2 Þ ¢ = + f x x ( ) 2 2 f x x f x ( ) ( ) ( ) ? = + Þ ¢ = 3 2 f x x x x ( ) ( ) = + = + + 3 6 9 2 2 Þ ¢ = + f x x ( ) 2 6 = + 2 1 ( ) x = + 2 3 ( ) x
  5. 6. f x x f x x ( ) ( ) ( ) ( ) = + Þ ¢ = + 1 2 1 2 f x x f x ( ) ( ) ( ) = + Þ ¢ = 7 2 f x x f x x ( ) ( ) ( ) ( ) = + Þ ¢ = + 3 2 3 2 2 7 ( ) x + f x x x x f x x x ( ) ( ) ( ) ( ) = + = + + Þ ¢ = + = + 7 14 7 2 14 2 7 2 2
  6. 7. f x x f x ( ) ( ) ( ) = + Þ ¢ = 2 3 2 2 2 3 ( ) x + 2 5 2 ( ) x + f x x x x x x f x x x ( ) ( ) ( )( ) ( ) ( ) = + = + + = + + Þ ¢ = + = + 2 3 2 3 2 3 4 12 9 8 12 4 2 3 2 2 Oops f x x x x f x x x ( ) ( ) ( ) ( ) = + = + + Þ ¢ = + = + 5 2 25 20 4 50 40 10 5 2 2 2 Oops again = + 2 2 2 3 . ( ) x = + 5 2 5 2 . ( ) x f x x f x ( ) ( ) ( ) = + Þ ¢ = 5 2 2
  7. 8. f x x f x ( ) ( ) ( ) = + Þ ¢ = 7 3 2 7 2 7 3 . ( ) x + f x x x x f x x x x ( ) ( ) ( ) ( ) . ( ) = + = + + Þ ¢ = + = + = + 7 3 49 42 9 98 42 14 7 3 7 2 7 3 2 2 f x x f x ( ) ( ) ( ) = + Þ ¢ = 2 1 3 2 3 2 1 2 . ( ) x + f x x x x x f x x x x x x x ( ) ( ) ( ) ( ) ( ) . ( ) = + = + + + Þ ¢ = + + = + + = + = + 2 1 8 12 6 1 24 24 6 6 4 4 1 6 2 1 2 3 2 1 3 3 2 2 2 2 2
  8. 9. f x x ( ) ( ) = + 2 2 1 f x x x x f x x x ( ) ( ) ( ) = + = + + Þ ¢ = + 2 2 4 2 3 1 2 1 4 4 = + = + 4 1 2 2 1 2 2 x x x x ( ) . ( )
  9. 10. f(x)= ( 2x + 3 ) 2 ⇒fʹ(x) = 2 . 2 (2x+3) 1 f(x) = ( 7x + 3 ) 2 ⇒fʹ(x) = 7 . 2 (7x + 3) 1 f(x) = ( 2x + 1 ) 3 ⇒ fʹ(x) = 2 . 3 (x + 1) 2 f(x) = ( x 2 + 1 ) 2 ⇒ fʹ(x) = 2x . 2 (x 2 + 1) 1 f(x) = ( x 3 + 3x 2 - 4x + 1 ) 7 ⇒ fʹ(x) = (3x 2 + 6x -4) . 7 (x 3 + 3x 2 -4x + 1) 6

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