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3.9 Exponential and Logarithmic Derivatives Wed Nov 12 Do Now Find the derivatives of: 1) 2)

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HW Review p.219 #9-20 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)

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HW Review p.219 #21-26 21) 22) 23) 24) 25) 26) 27) 3sec^2 4x (sec4xtan4x) (4) 28) e^sin2x (cos2x) (2)

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Exponential + Logarithmic Functions Logarithmic and exponential functions are among the most common functions encountered in applications. Population curves consist of logarithmic functions, particularly the natural logarithm. Growth/Decay, business applications use exponential functions

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Thm- For any constant b > 0, Thm- In particular,

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Derivative of Natural Log To determine the derivative of the natural logarithm, let’s take a look at the graph of lnx and its slopes

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Derivative of ln x cont’d Thm- For x > 0,

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Example: Find the derivative of f(x) = x ln x and g(x) = x 10^x

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Closure Hand in: Find the derivative of the following functions 1) 2) HW: p.187 #1-19 odds, 25-31 odds

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Logarithmic Differentiation Thurs Nov 13 Do Now Find the derivative 1) 2)

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HW Review: p.187 #1-19 25-31 1) lnx + 119) 3) (2/x) lnx25) y = 36ln6(x-2) + 36 5)27) y = 3^20 ln3(t-2) + 3^18 7)29) y = 5^-1 9)31) y = -1(t-1) + ln4 11) 13) 15) 17)

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Logarithmic Differentiation Logarithmic Differentiation can be used in place of several product/quotient rules Ex:

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Logarithmic Differentiation 1) Take ln of both sides 2) Use log rules to separate each factor 3) Differentiation both sides (chain rule) 4) Multiply by f(x) (original)

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Ex Use log differentiation

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Ex 2 Differentiate using log dif.

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Closure Find the derivative using logarithmic differentiation HW: p.187 #37-43 odds

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Other base Log Derivatives Fri Nov 14 Do Now Find the derivative 1) 2)

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HW Review: p.187 #37-43 37) 2x + 14 39) 3x^2 - 12x + 79 41) 43)

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Other Base Logarithms We can calculate the derivative of other base logs by using the change-of-base formula using ln x

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Ex Find the derivative of

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You try Find the derivatives 1) 2)

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Closure Journal Entry: Write about the derivatives of logarithms. How can we calculate them? How can they be useful when we have complicated functions? HW: p.187 #21-24

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