4.4 and 4.5: Derivatives of Exponential and Log Functions

Review Properties of Logs and Exponential Function

Inverse:logax = y ay = x

lnx = y ey=x

Other properties:ln ax = xlnalogax= xloga

Properties of Logs and Exponential Functions cont.

logaax=x

lnex = x =xelnx = xChange of base: logax =

a

x

b

b

log

log

xaa log

Derivative of ex:

Derivative of ax:

xx eedx

d )(')()( xgee

dx

d xgxg

xx aaadx

d)(ln )(')(ln )()( xgaaa

dx

d xgxg

Examples: Find the derivative.

1.

2.

3.

4.

xey 6

xey 5

12

2)( xxf

xxg 5)(

Find the derivative.

1. 2.

3. 4.

xexey 2

xy cot3

xey 2sin

24 3 xey x

1. 32

3)( tettg 2. t

t

et

ety

3

22

When does the tangent line to the graph of y = 2t -3 have a slope of 21?

Derivatives of Logarithmic Functions

xa

xdx

da

ln

1log )('

)(ln

1)(log xg

xgaxg

dx

da

x

xdx

d 1ln )('

)(

1)(ln xg

xgxg

dx

d

Find the derivative.

1. 2.

3. 4.

2log xy )1ln( 2 xy

xxy 4log 32 3log)( 5 xxf

Examples: Find the derivative.

1. 2. )2ln(3

xey x 2ln)( xxexf

An absolute value inside of a logarithm has no effect on the derivative, other than make the result valid for more x values.

(see p. 287)

Example: Find the derivative.

xxy 28ln 3