03Derivatives of Logarithmic and Exponential Functions

03Derivatives of Logarithmic and Exponential Functions - ,...

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D ERIVATIVES OF L OGARITHMIC AND E XPONENTIAL F UNCTIONS Recall y x b y b x log = = R x b b y , 1 , 0 , Natural log: x x e ln log = To differentiate log and exponential functions 0 x when e b = , 1 ln = e , so we get: 0 x Ex . Find [ ] ) 1 ln( 2 + x dx d . Ex . Differentiate + x x x 1 sin ln 2 Use log properties to simplify. Consider 4 2 3 2 ) 1 ( ) 14 7 ( x x x y + - = nasty to do directly, so take ln of both sides. [ ] b x e x x dx d b b ln 1 log 1 log = = [ ] x x dx d 1 ln =
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) 1 ln( 4 ) 14 7 ln( 3 1 ln 2 ln 2 x x x y - - - + = Now differentiate both sides with respect to x : Implicitly 2 1 8 14 7 3 7 2 1 x x x x dx dy y + - - + = Solve for dx dy : + - - + + - = 2 4 2 3 2 1 8 6 3 1 2 ) 1 ( ) 14 7 ( x x x x x x dx dy Differentiating Exponential Functions Take y x b log = and differentiate implicitly. dx dy b y ln 1 1 = Solve for dx dy
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Unformatted text preview: , replace y by x b b b b y dx dy x ln ln = = so: b b b dx d x x ln = when e b = => [ ] x x e e dx d = Ex . Find a) [ ] x dx d sin 2 b) [ ] x e dx d 2-c) [ ] 3 x e dx d Three Cases: 1. ( 29 ln x x dx d = variable in exponent 2. ( 29 x dx d = 1-x variable in base 3. ( 29 x x dx d variable in exponent and base, take ln of both sides and differentiate implicitly Assign: p.267 #1-27, 29-33, 35, 37, 39, 43, 45, 47; Worksheets (3)...
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03Derivatives of Logarithmic and Exponential Functions - ,...

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