Techniques in Differentiation

Techniques in Differentiation - Contents 3 Techniques of...

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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
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CONTENTS 2
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Chapter 3 Techniques of Differentiation 3.1 Differentiating Functions Involving the Power Rule, Constants and the Sums of Differentiable Functions For questions numbered 1–23, compute the indicated derivative. The letters a, b, c, α, β, γ , 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 0 4. y = ax + b c ; y 0 y 0 = a c 5. y = 1 + x + x 2 2! + x 3 3! + · · · + x n n ! ; y 0 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 ) = α + ( α + β ) t + ( α + β + γ ) t 2 ; G 0 (2) [ G 0 (2) = 5 α + 5 β + 4 γ ] 8. y ( x ) = ax 2 + bx x ; y 0 ( x ) 9. F ( t ) = a 2 +2 abt + b 2 t 2 a 2 +2 ab + b 2 ; F 0 (1) 10. y = a 2 +2 abx + b 2 x 2 a + bx ; y 0 [ y 0 = b ] 11. y = x 2 - 1 x +1 ; y 0 12. y = ax 1 2 + bx 3 2 + cx 5 2 x ; y 0 13. f ( x ) = 5 x 101 + 17 x 2 + 8; f 0 ( x ) [ f 0 ( 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 0 (2) [ f 0 (2) = 1] 17. f ( x ) = x + | x - 3 | ; f 0 (2)
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45 3.2 Differentiating Functions Involving Products, Powers and Quotients 18. f ( x ) = | x | x ; f 0 (1) and f 0 ( - 1) 19. f ( x ) = x + | x 2 - 2 | ; f 0 (1) [ f 0 (1) = - 1] 20. f ( x ) = x + | x 2 - 2 | ; f 0 (3) 21. It is true that sin x = x - x 3 3! + x 5 5! - · · · + ( - 1) n x 2 n +1 (2 n +1)! + · · · cos x = 1 - x 2 2! + x 4 4! - · · · + ( - 1) n x 2 n (2 n )! + · · · These are infinite series representations of sin and cos. By differentiating the series term by term, find formulae for D x sin x and D x cos x . 22. F ( x 1 2 ) = ax 3 2 + bx 5 2 + cx 5 ; D x 1 2 F ( x 1 2 ) h D x 1 2 F ( x 1 2 ) = 3 ax + 5 bx 2 + 10 cx 9 2 i 23. (a) F ( x 2 ) = x 6 + 3 x 2 + 1; D x 2 F ( x 2 ) (b) F ( x ) = x 2 + x + 2; D x F ( x ) (c) F ( x + 1) = x 2 + 2 x + 2; D ( x +1) F ( x + 1) (d) F ( a - x ) = a 2 - 2 ax + x 2 - a - x ; D ( a - x ) F ( a - x ) 24. Use the definition of the derivative to prove that D x ( af ( x ) - bg ( x ) ) = aD x f ( x ) - bD x g ( x ). 3.2 Differentiating Functions Involving Products, Powers and Quo- tients For questions numbered 25–53, compute the derivative. Answers should be simplified and in a factored form wherever possible. 25. y = ( a + bx 2 ) 3 ; y 0 y 0 = 6 bx ( a + bx 2 ) 2 26. y = ( x 2 + 2 x )(3 x + 1); dy dx 27. y = ( x 3 + 6 x 2 - 2 x + 1)( x 2 + 3 x - 5); y 0 28. y = (3 + x 3 )(4 + x 2 ); D x y D x y = x (5 x 3 + 12 x + 16) 29. z = ( a - x )( a 2 + ax + x 2 ); dz dx 30. W = ( a + bx )( a 4 - a 3 bx + a 2 b 2 x 2 - ab 3 x 3 + b 4 x 4 ); W 0 31. f ( x ) = a - x x ; f 0 ( x ) f 0 ( x ) = - a x 2 32. f ( x ) = x a - x ; f ( x ) 33. y = x - a x + a ; D x y 34. y = x + a x - a ; D x y h D x y = - 29 ( x - 9) 2 i 35. F ( t ) = 1+ t 1+ t 2 ; F 0 ( t )
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Techniques of Differentiation 46 36. F ( t ) = t 2 +2 t +1 ( t +1) 2 ; F 0 ( t ) 37. f ( x ) = 2 x 2 - 3 x x 2 +3 ; f 0 ( x ) h f 0 ( x ) = 3 x 2 +12 x - 9 ( x 2 +3) 2 i 38. T ( t 3 - 3 t + 3) 3 ; dT dt 39. y = 2 x 2 - 3 x +4 x ; y 0 40. y = 3 x - 2 2 x +3 ; y 0 41. φ = 2 θ +3 3 θ +2 ; φ 0 42. y = x x 2 +1 ; D x y 43. φ = (2 x 3 + x 2 - 4 x - 1) 4 ; φ 0 φ 0 = 8(3 x 2 + x - 2)(2 x 3 + x 2 - 4 x - 1) 3 44. ψ ( φ ) = ( θ +3) 2 θ +2 ; ψ 0 ( φ ) 45. τ ( θ ) = (1 + θ 2 )(1 + θ 3 ); τ 0 ( θ ) 46. y = 1+
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