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4.4 - MATH1081 Monday,February21 Chapter4Sections4&5...

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MATH 1081 Monday, February 21 Chapter 4 – Sections 4 & 5 DERIVATIVES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS Homework #6 (due 2/28): Section 4.4 #32, 38, 58 Section 4.5 #42, 58 Section 5.1 #26, 48, 52 Clicker Check-in : Choose any letter to check in now.
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Today we will look at our last two rules of differentiation: the derivative of an exponential function and the derivative of a logarithmic function. An exponential function is a function of the form f x ( ) = b x where b > 0 and b 1 . A logarithmic function is a function of the form f x ( ) = log b x ( ) where b > 0 and b 1 . The “natural” base that is the number e 2.71828 . In the case of the logarithmic function, log e x ( ) = ln x ( ) .
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Let’s start with f x ( ) = e x and the definition of derivative. d dx e x = lim h 0 e x + h e x h = lim h 0 e x e h e x h = lim h 0 e x e h 1 ( ) h It turns out that (although the proof won’t be done here) the limit lim h 0 e h 1 h = 1 . So, d dx e x = e x .
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The function f x ( ) = e x is its own derivative.
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