lecture24

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

This preview shows pages 1–4. Sign up to view the full content.

§ 4.5 The Derivative of ln x Math 121 Lecture 24 ! Derivatives for Natural Logarithms ! Differentiating Logarithmic Expressions

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 ]
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Set the numerator equal to 0.
This is the end of the preview. Sign up to access the rest of the document.

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, let’s 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)....
View Full Document

## This document was uploaded on 09/30/2010.

### Page1 / 4

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online