Chapter 3 Section 09

Chapter 3 Section 09 - 2 1 2 1 2 1 2 1 2 1 2 1 1 1 cot 1 1...

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3.9 Derivatives of Inverse Functions Page 187
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Objectives: 1. Find the derivative of the inverse of a function, without actually finding the inverse function. 2. Find the derivatives of the inverse trig functions.
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Theorem Derivative of the Inverse: Assume that f(x) is differentiable and one-to-one with inverse g(x). If b belongs to the domain of g(x) and f’(g(b)) 0, then g’(b) exists and )) ( ( ' 1 ) ( ' b g f b g =
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Page 191, #12. Calculate g(-2) and g’(-2) where g is the inverse of f(x) = 4x 3 – 2x. g(-2) f(x) = -2 4x 3 – 2x = -2 4x 3 – 2x + 2 = 0 2(x + 1)(2x 2 – 2x + 1) = 0 x = -1
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Page 191, #12. Calculate g(-2) and g’(-2) where g is the inverse of f(x) = 4x 3 – 2x. ( 29 10 1 2 10 2 12 2 4 1 1 2 2 1 2 1 2 3 = - = - = - - = - - = - = ) ( ' (-1) f'   ; ) ( ' ) ( ' )) ( ( ' ) ( ' )) ( ( ' ) ( ' g x x x dx d f g g f g b g f b g
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Compute the formula for the derivative of sin -1 (x). [ ] 2 1 2 1 1 1 1 - 1 - 1 1 ) ( sin , 1 1 )) ( cos(sin 1 ) ( sin )) ( cos(sin )) ( ( ' )) ( ( ' 1 ) ( ' (x). sin g(x) (x) f and cos(x), (x) f' then sin(x), f(x) Let x x dx d Therefore x x x dx d x x g f x g f x g - = - = = = = = = = = - - - -
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2 1 1 1 1 )) ( cos(sin , 1 sin ) ( sin )) ( cos(sin x x Therefore x then x Let x - = = = - - - θ 1 x θ √(1-x 2 )
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Inverse trig function derivatives [ ] [ ] [ ]
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Unformatted text preview: [ ] [ ] [ ] 2 1 2 1 2 1 2 1 2 1 2 1 1 1 ) ( cot 1 1 ) ( sec 1 1 ) ( csc 1 1 ) ( tan 1 1 ) ( cos 1 1 ) ( sin x x dx d x x x dx d x x x dx d x x dx d x x dx d x x dx d +-=-=--= + =--=-=------4 5 5 4 1 = = = = = = = 25 16 1 25 9-1 1 5 3-1 1 ) 5 3 ( f' x-1 1 (x) f' (x). sin f(x) if ) 5 3 ( f' Find #18. 191, page 2 2 1-( 29 ( 29 4 2 2 2 2 2 2 1--1 2x-2 x-1 1-x x-1 1-(x) ' f ). (x cos f(x) if ) (x ' f Find #24. 191, page x x dx d = ⋅ = ⋅ = = ) ( tan 1 ) ( tan 1 1 ) ( tan ) ( tan (x) ' f (x). xtan f(x) if ) (x ' f Find #26. 191, page 1 2 1 2 1 1 1-x x x x x x x dx d x x dx d x----+ + = + + ⋅ = ⋅ + ⋅ = = ( 29 ( 29 x x x x x x e e e e e dx d e ⋅-= ⋅-= ⋅-= = 2 2 2 x 1 1 1 1 1 1 (x) ' f ). arcsin(e f(x) if ) (x ' f Find #28. 191, page...
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Chapter 3 Section 09 - 2 1 2 1 2 1 2 1 2 1 2 1 1 1 cot 1 1...

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