3.9 Exponential and Logarithmic Derivatives Wed Nov 12 Do Now Find the derivatives of: 1) 2)
HW Review p.219 #9-20 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
HW Review p.219 # ) 22) 23) 24) 25) 26) 27) 3sec^2 4x (sec4xtan4x) (4) 28) e^sin2x (cos2x) (2)
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
Thm- For any constant b > 0, Thm- In particular,
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
Derivative of ln x cont’d Thm- For x > 0,
Example: Find the derivative of f(x) = x ln x and g(x) = x 10^x
Closure Hand in: Find the derivative of the following functions 1) 2) HW: p.187 #1-19 odds, odds
Logarithmic Differentiation Thurs Nov 13 Do Now Find the derivative 1) 2)
HW Review: p.187 # ) lnx + 119) 3) (2/x) lnx25) y = 36ln6(x-2) )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)
Logarithmic Differentiation Logarithmic Differentiation can be used in place of several product/quotient rules Ex:
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)
Ex Use log differentiation
Ex 2 Differentiate using log dif.
Closure Find the derivative using logarithmic differentiation HW: p.187 #37-43 odds
Other base Log Derivatives Fri Nov 14 Do Now Find the derivative 1) 2)
HW Review: p.187 # ) 2x ) 3x^2 - 12x ) 43)
Other Base Logarithms We can calculate the derivative of other base logs by using the change-of-base formula using ln x
Ex Find the derivative of
You try Find the derivatives 1) 2)
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