Section 10.2 Derivatives of Logarithmic and Exponential Functions:

Introduction: We have looked at many derivatives, but up until now we haven’t looked at exponential or logarithmic functions and their derivatives. In this section, we will look at the derivative of logarithmic and exponential functions.

In this section we will discuss:

•Derivative of the natural logarithm function •Differentiating the natural logarithm function with the chain rule •Derivative of the exponential function with base e •Derivative of the exponential functions with the chain rule •Applications

Derivative of Natural Logarithm Function: The Derivative of y =ln x: fx()= ln x fx' = 1 () x

Review of Log Laws The following three log laws come in handy when looking at complicated functions involving logarithms:

5. ln()M ⋅=NM ln + ln N ⎛⎞M 6.ln⎜⎟=− lnMN ln ⎝⎠N 8. lnMpMp = ln

Examples of Derivatives Involving the Natural Logarithm Function Find the derivative for the following functions: 4 1.fx()= ln x

3.fx()=⋅ 12 x2 ln x

Differentiating the Natural Logarithmic Function with the Chain Rule Let’s look again at our first example, without using the log laws. We could use the chain rule. fx()= ln ( inside ) 1 fx'()=⋅ derivofinside inside

fx()= ln x4

Example 2: gx()=+ln ( 3 x 4 )

4 Example 3: gx()= (ln x )

Compare these. Think about where the INSIDE is in each question and where the OUTSIDE is. See how these two differ?

More Examples of Differentiating Logarithmic Functions with the Chain Rule: 33 Example 4: gx()=⋅4ln3 x( x)

42xx3 −+ Example 5: gx()= ln()x + 7

Derivative of the Exponential Function: Derivative of y = ex

Examples of Derivatives of the Exponential Function: 1. fx( ) = xe3 x fx'3()=+ xe32xx e() x

ex + 6 2. gx()= xx2 − 9 ()()()xxee2 −−+96xx()2 x−9 ' = gx() 2 ()xx2 − 9

3. fx()=+( ex 1) ln x

Differentiating Exponential Functions Involving the Chain Rule: The first thing you need to realize when you see a function that looks like this: gx()= e69x− you need to realize that the exponent is the “inside” function. If you are really confused about what the inside function is, ask yourself “if I were to plug a number in for x, what is the first thing I would do?” In this case, you would have to plug it in to 6x -9 and then take e and raise it to that power, so the first thing is the “inside” 69x− and the e is the “outside”. The derivative is the following: gx'6()= e ()

Differentiating Exponential Functions Involving the Chain Rule: To generalize: If fx() gx()= e fx() gx''()= e fx()

Examples of Differentiating Exponential Functions Involving the Chain Rule: Find the derivative of the following: Example 1: 3 gx()= ex Rewrite 1 3 gx()= ex

1 3 −2 gx' =⋅ ex 1 x 3 () ( 3 )

Example 2: e2+x gx()= e2−x 22−+xx 22 +− xx ()()ee()11−−()() ee() ' = gx() 2 ()e2−x

2x−1 Example 3: gx()=⋅+() eln() 6 x 4

Applications: Example 1: The percentage of alcohol in a person's blood stream after drinking 8 fluid ounces −04. t of 100 proof whiskey is approximated by At()=≤023 . te , 0 t 12 a) What is the percentage of alcohol in a person’s bloodstream after 30 minutes? After 8 hours? b) How fast is the percentage of alcohol in a person’s bloodstream changing after 30 minutes? After 8 hours?

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