Examples of How to Find the Derivative Using The Power Rule and Product Rule

# Examples of How to Find the Derivative Using The Power Rule and Product Rule

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Examples of How to Find the Derivative Using The Power Rule, Product Rule, Quotient Rule and Chain Rule Power Rule Examples: 1. y = 3x 4 -2x + 1 y’ = 12x 3 -2 2. y = 3 x y = x 3 1 y’ = 3 2 3 1 - x 3. y = 12x 5 + 3x 4 -5x y’ = 60x 4 + 12x 3 -5 4. y = x + x 1 y = 2 1 2 1 - + x x y’ = 2 3 2 1 2 1 2 1 - - - + x x 5. v(r) = π 3 4 r 3 v’(r) = 4 π r 2 6. f(x) = 12 f’(x) = 0 7. f(x) = 3x 4 (x 2 -2x + 1) f(x) = 3x 6 -6x 5 +3x 4 f’(x) = 18x 5 -30x 4 +12x 3 8. f(t) = 4 4 4 4 1 t t - f’(t) = t t 4 5 3 1 1 - +

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Product Rule Examples: 9. f(x) = 4x(3x 2 + x ) f’(x) = 4x (6x + 2 1 2 1 - x ) + 4 (3x 2 + x ) 10. y = (2x 2 + 5) (4x – 1 ) y’ = (2x 2 + 5) (4) + (4x) (4x – 1 ) 11. t(x) = (2x 4 – 1)(5x 3 + 6x) t’(x) = (2x 4 – 1)(15x 2 + 6) + (8x 3 ) (5x 3 + 6x) 12. y = ) 3 )( 1 ( 3 x x x + y = ) 3 )( ( 2 1 2 1 3 1 x x x - + y’ = ) 3 )( 2 1 3 1 ( ) 2 3 )( 1 ( 2 3 3 2 2 1 3 x x x x x x - - - - + + 13. f(x) = (4x 2 )(3x + 5)(6x 2 + x) f”(x) = (4x 2 )[ (3x + 5)(12x + 1) + (3)( 6x 2 + x)] + 8x (3x + 5)(6x 2 + x) 14. y = (3x 2 – x -1 )(5x)(2x -2 ) y’ = (3x 2 – x -1 )[ (5x)(-4x -3 ) + 5(2x -2 )] + (6x + 1x -2 )( 5x)(2x -2 ) 15. y = 12 (x 5 – 3x) y’ = 12 (5x 4 – 3 ) Examples of Quotient and Chain Rules 16. f(x) = 1 - x x f’(x) = ) 1 ( 2 ) 1 )( ( ) 1 )( 1 ( - - - x x x 17. f(y) = 4 3 1 2 + + y y f’(y) = ) 4 3 ( 2 ) 3 )( 1 2 ( ) 2 )( 4 3 ( + + - + y y y
18. g(x) = 2 1 5 x x + g’(x) = 2 2 2 ) 1
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