3.9 Exponential and Logarithmic Derivatives

Wed Nov 12Do 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 #21-26• 21)• 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, 25-31 odds

Logarithmic DifferentiationThurs Nov 13

• Do Now

• Find the derivative

• 1)

• 2)

HW Review: p.187 #1-19 25-31

• 1) lnx + 1 19)• 3) (2/x) lnx 25) 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)

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 DerivativesFri Nov 14

• Do Now

• Find the derivative

• 1)

• 2)

HW Review: p.187 #37-43

• 37) 2x + 14

• 39) 3x^2 - 12x + 79

• 41)

• 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