Techniques in Differentiation

Techniques in Differentiation - Contents 3 Techniques of...

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Unformatted text preview: Contents 3 Techniques of Differentiation 44 3.1 Differentiating Functions Involving the Power Rule, Constants and the Sums of Differentiable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2 Differentiating Functions Involving Products, Powers and Quotients . . . . . . . . . . . . . . 45 3.3 Differentiating the Composite of Functions Using the Chain Rule . . . . . . . . . . . . . . . . 47 3.4 Differentiating Rational Powers of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.5 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.6 Derivatives of Logarithmic and Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 52 3.7 Higher Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.8 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.9 Derivatives of Functions of the Form f ( x ) = g ( x ) h ( x ) . . . . . . . . . . . . . . . . . . . . . . . 59 3.10 Logarithmic Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.11 Tangent Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 CONTENTS 2 Chapter 3 Techniques of Differentiation 3.1 Differentiating Functions Involving the Power Rule, Constants and the Sums of Differentiable Functions For questions numbered 123, compute the indicated derivative. The letters a,b,c,,, , represent con- stants. Answers should be simplified and in a factored form wherever possible. 1. y = 3 x 5- 2 x 3 ; D x y D x y = 3 x 2 (5 x 2- 2) 2. z = 3 ax 2 + 2 bx + c ; dz dx 3. y = ( 2) 3- ( 2) 5 ; y 4. y = ax + b c ; y y = a c 5. y = 1 + x + x 2 2! + x 3 3! + + x n n ! ; y 6. y = ( 7 2 x 3- 3 x 2 + 1 3 x- 1 )- ( 2 x 4 + 2 x 3- 1 2 x 2- 2 ) ; D x y 7. G ( t ) = + ( + ) t + ( + + ) t 2 ; G (2) [ G (2) = 5 + 5 + 4 ] 8. y ( x ) = ax 2 + bx x ; y ( x ) 9. F ( t ) = a 2 +2 abt + b 2 t 2 a 2 +2 ab + b 2 ; F (1) 10. y = a 2 +2 abx + b 2 x 2 a + bx ; y [ y = b ] 11. y = x 2- 1 x +1 ; y 12. y = ax 1 2 + bx 3 2 + cx 5 2 x ; y 13. f ( x ) = 5 x 101 + 17 x 2 + 8; f ( x ) [ f ( x ) = 505 x 100 + 34 x ] 14. y = ( a + bx 2 ) 3 ; D x y 15. y = x 3- a 3 x- a ; D x y 16. f ( x ) = | x | ; f (2) [ f (2) = 1] 17. f ( x ) = x + | x- 3 | ; f (2) 45 3.2 Differentiating Functions Involving Products, Powers and Quotients 18. f ( x ) = | x | x ; f (1) and f (- 1) 19. f ( x ) = x + | x 2- 2 | ; f (1) [ f (1) =- 1] 20. f ( x ) = x + | x 2- 2 | ; f (3) 21. It is true that sin x = x- x 3 3!...
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This note was uploaded on 09/22/2011 for the course ECON 101 taught by Professor Mr.tull during the Spring '11 term at De La Salle University.

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Techniques in Differentiation - Contents 3 Techniques of...

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