3_6 - y = ln F ( x ), using the Laws of Logarithms to...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Section 3.6: Derivatives of Logarithmic Functions Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu) Logarithmic Functions The derivative of exponential functions The derivative of the exponential function F ( x ) = a x is F 0 ( x ) = a x ln a . Use the Chain Rule (Section 3.4), we have the general formula: [ a g ( x ) ] 0 = a g ( x ) (ln a ) g 0 ( x ) The derivative of logarithmic functions Use the method of implicit differentiation in Section 3.5, we can find the derivative of the logarithmic function: F ( x ) = log a x F 0 ( x ) = 1 x ln a (check the proof on page 215, textbook). The derivative of the natural logarithmic function: F ( x ) = ln x F 0 ( x ) = 1 x Remark : log a a = 1 ln e = log e e = 1 Use the Chain Rule, we have the general formula: [ln g ( x )] 0 = g 0 ( x ) g ( x ) Example 1 of Section 3.6 (textbook): Example 2 of Section 3.6 (textbook): Example 3 of Section 3.6 (textbook): Example 4 of Section 3.6 (textbook): Example 5 of Section 3.6 (textbook): Example 6 of Section 3.6 (textbook): The method of logarithmic differentiation Sometimes, the calculation of the derivative of a function y = F ( x ) can be simplified by: taking logarithms ln
Background image of page 1

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

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: y = ln F ( x ), using the Laws of Logarithms to simplify the new equation, 1 dierentiating the equation implicitly with respect to x , solving the resulting equation for y . The Laws of Logarithms : Given a > , a 6 = 1, x and y are positive numbers, then 1. log a ( xy ) = log a x + log a y 2. log a ( x y ) = log a x-log a y 3. log a ( x r ) = r log a x Example 7 of Section 3.6 (textbook): Example 8 of Section 3.6 (textbook): The Number e as a Limit In Section 3.1, we learned that: if f ( x ) = a x ( a > , a 6 = 1) then e is the number such that f (0) = lim h e h-1 h = 1. Reminder : The geometric meaning of the slopes f (0) of the exponential function f ( x ) = a x In this section, e can be rigorously dened as e = lim x (1 + x ) 1 x (check the proof on page 219 of the textbook). From the above graph and table, e 2 . 7182818. 2...
View Full Document

This note was uploaded on 05/23/2011 for the course MAC 2311 taught by Professor Noohi during the Fall '08 term at FSU.

Page1 / 2

3_6 - y = ln F ( x ), using the Laws of Logarithms to...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online