Section 3.9

DERIVATIVES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

If you recall, the number e is important in manyinstances of exponential growth:

1lim 1x

xe

x

Find the following important limit using graphsand/or tables:

0

1limh

h

eh

1

Derivative of xe

1xe

x xd e edx

0

limx h x

x

h

d e eedx h

0lim

x h x

h

e e eh

0

1limh

x

h

eeh

0

1limh

x

h

eeh

The limit we just figured!

Definition of thederivative!!!

The derivative of this function is itself!!!

Derivative of xa

ln lnxax x aa e e

lnx x ad da edx dx

ln lnx a de x adx

ln lnx ae a lnxa a

Given a positive base that is not one, we can use a propertyof logarithms to write in terms of :xa xe

Derivative of ln xlny x

yd de xdx dx

ye x 1y

dydx e

1y dyedx

1dy

dx xImp. Diff. Su

bstit

ution

!

Derivative of loga xFirst off, how am I able to express in thefollowing way??? lnlog

lnaxxa

1 lnln

d xa dx

lnloglna

d d xxdx dx a

1 1ln a x

COB Formula!

1lnx a

loga x

Summary of the New Rules(keeping in mind the Chain Rule and any variable restrictions)

u ud due edx dx

lnu ud dua a adx dx

0, 1a a

1lnd duudx u dx

0u

1loglna

d duudx u a dx

0, 1a a

Now we can realize the FULL POWERof the Power Rule……………observe:

lnn n xx e

ln lnn x de n xdx

lnn n xd dx edx dx

lnn x nex

1nnx

Start by writing x with any real power as a power of e…

1nx nx

Power Rule for Arbitrary Real Powers

1n nd duu nudx dx

If u is a positive differentiable function of x and n isany real number, then is a differentiable functionof x, and

The power rule works for not only integers, not only rational numbers, but any real numbers!!!

nu

Quality Practice Problems

3 3y x Find : 3 43 3dy xdx

dydx

34 xy e 312 xdy edx

Find :dydx

4 15 xy 4 14 5 ln 5xdydx

Find :dydx

Quality Practice Problems

Find :dydx

3lny x 23

1 3dy xdx x

3 , 0xx

1 1ln 5 2

dydx x x

1 , 02 ln 5

xx

5logy xFind :dydx

Quality Practice Problems

Find :dydx

xy xHow do we differentiate a function when both the base and exponent

contain the variable???

Use Logarithmic Differentiation:1. Take the natural logarithm of both sides of the equation

2. Use the properties of logarithms to simplify the equation

3. Differentiate (sometimes implicitly!) the simplified equation

Quality Practice Problems

Find :dydx

xy x

ln lnd dy x xdx dx

ln 1dy y xdx

ln 1xdy x xdx

ln lny x x

ln ln xy x1 1 1 lndy x xy dx x

Quality Practice ProblemsFind using logarithmic differentiation:

dydx

2

2 2

1

xxy

x

2

2 2ln ln

1

xxy

x

21ln ln 2 ln 2 ln 12

y x x x

Differentiate: 2

1 1 1 12 ln 2 22 2 1

dy xy dx x x

Quality Practice ProblemsFind using logarithmic differentiation:

dydx

2

1 1 1 12 ln 2 22 2 1

dy xy dx x x

2

1 ln 21

dy xydx x x

Substitute: 22

2 2 1 ln 211

xx xx xx

Quality Practice ProblemsA line with slope m passes through the origin and is tangentto the graph of . What is the value of m? What does the graph look like?

lny x

, lna a1ma

The slope of the curve:

ln 0 ln0

a ama a

The slope of the line:

Now, let’s set them equal…

1ma

lny x

0,0

Quality Practice ProblemsA line with slope m passes through the origin and is tangentto the graph of . What is the value of m? What does the graph look like?

lny x

, lna a1ma

lny x

ln 1aa a

ln 1a 1a e

So, our slope: 1me

0.368 0,0