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Chapter 3 Section 10

Chapter 3 Section 10 - 3.10 Derivatives of General...

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1 3.10 Derivatives of General Exponential and Log Functions page 192
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2 Objectives 1. Find derivatives of general exponential functions, a x . 2. Find derivatives of functions involving the log function. 4. Find derivatives using log differentiation. 5. Find derivatives of the hyperbolic functions.
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3 From sec 2: Exponential Functions: D x [a x ] ( 29 ( 29 ) 0 ( ' ) ( ' , ) 0 ( ' 1 , 0 1 lim 1 lim lim lim ) ( ' 0 0 0 0 0 f a x f so f h a a h a h as h a a h a a h a a a h a a x f x h h h h x h x h x h x h x h x h = - = - - = - = - = - = + Which means, the derivative of an exponential is proportional to the function itself.
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4 General Exponential Rule: D x [a x ] [ ] ( 29 [ ] [ ] [ ] ) ln( ) ln( ) ln( ) ln( ) ln( ) ln( a a a dx d a a a e e dx d e dx d a dx d x x x x a x a x a x = = = = =
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5 ( 29 ) 1 2 )( 5 ln( 5 ) 5 ln( 5 5 #18. 197, page 2 2 2 2 - = - = = - - - x dx dy x x dx d dx dy y x x x x x x
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6 Page 198, #26 Find the equation of the line tangent to the curve f(x) = π 3x+9 at x = 1. y - f(a) = f ’(a)(x - a) x = 1 f(1) = π 12 (1, π 12 ) y – π 12 = f ’(1)(x - 1) f ’(x) = π 3x+9 ln( π )(3) f ’(1) = 3 π 12 ln( π ) y – π 12 = 3 π 12 ln( π ) (x - 1)
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7 -4 -3 -2 -1 1 2
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