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# 3_6 - y = ln F x • using the Laws of Logarithms to...

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Section 3.6: Derivatives of Logarithmic Functions Instructor: Ms. Hoa Nguyen Logarithmic Functions The derivative of exponential functions The derivative of the exponential function F ( x ) = a x is F ( x ) = a x ln a . Use the Chain Rule (Section 3.4), we have the general formula: [ a g ( x ) ] = a g ( x ) (ln a ) g ( 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 ( x ) = 1 x ln a (check the proof on page 215, textbook). The derivative of the natural logarithmic function: F ( x ) = ln x F ( 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 )] = g ( 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:

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Unformatted text preview: y = ln F ( x ), • using the Laws of Logarithms to simplify the new equation, 1 • diﬀerentiating 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 deﬁned 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...
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3_6 - y = ln F x • using the Laws of Logarithms to...

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