lecture24 - The derivative will equal 0 when the numerator...

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§ 4.5 The Derivative of ln x Math 121 Lecture 24 ! Derivatives for Natural Logarithms ! Differentiating Logarithmic Expressions
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Differentiate. e [(ln x ) 2 ] This is the given expression. Differentiate. d dx e [(ln x ) 2 ] Use the chain rule. = e [(ln x ) 2 ] " d dx [(ln x ) 2 ] Use the power rule = e [(ln x ) 2 ] " 2 " ln x " d dx (ln x ) Finish. = e [(ln x ) 2 ] " 2 " ln x " 1 x = 2 e [(ln x ) 2 ] ln x x e [(ln x ) 2 ]
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Differentiate. ln e 2 x + 1 ( ) [ ] 2 This is the given expression. ln e 2 x + 1 ( ) [ ] 2 Differentiate. d dx ln e 2 x + 1 ( ) [ ] 2 { } Use the power rule. 2ln e 2 x + 1 ( ) " d dx ln e 2 x + 1 ( ) [ ] Differentiate ln[ g ( x )]. 2ln e 2 x + 1 ( ) " 1 e 2 x + 1 " d dx e 2 x + 1 ( ) Finish. 2ln e 2 x + 1 ( ) " 1 e 2 x + 1 " 2 e 2 x The function has a relative extreme point for x > 0. Find the coordinates of the point. Is it a relative maximum point? This is the given function. Use the quotient rule to differentiate. Simplify. Set the derivative equal to 0.
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Set the numerator equal to 0.
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Unformatted text preview: The derivative will equal 0 when the numerator equals 0 and the denominator does not equal 0. Write in exponential form. To determine whether the function has a relative maximum at x = 1, lets use the second derivative. This is the first derivative. Differentiate. " " f x ( ) = # x 2 $ 1 x # ln x $ # 2 x ( ) # x 2 ( ) 2 Simplify. Factor and cancel. Evaluate the second derivative at x = 1. Since the value of the second derivative is negative at x = 1, the function is concave down at x = 1. Therefore, the function does indeed have a relative maximum at x = 1. To find the y-coordinate of this point So, the relative maximum occurs at (1, 1)....
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lecture24 - The derivative will equal 0 when the numerator...

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