Section 3.9

DERIVATIVES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

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

1lim 1

x

xe

x

Find the following important limit using graphsand/or tables:

0

1lim

h

h

e

h

1

Derivative of xe

1xe

x xde e

dx

0

limx h x

x

h

d e ee

dx h

0lim

x h x

h

e e e

h

0

1lim

hx

h

ee

h

0

1lim

hx

h

ee

h

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 e

dx dx ln lnx a d

e 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 x

lny x

yd de x

dx dx

ye x 1y

dy

dx e

1y dye

dx

1dy

dx xImp. Diff. Su

bstit

ution

!

Derivative of loga x

First off, how am I able to express in thefollowing way???

lnlog

lna

xx

a

1ln

ln

dx

a dx

lnlog

lna

d d xx

dx dx a

1 1

ln a x

COB Formula!

1

lnx a

loga x

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

u ud due e

dx dx lnu ud du

a a adx dx

0, 1a a

1ln

d duu

dx u dx

0u

1log

lna

d duu

dx 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 x

dx lnn n xd d

x edx dx

lnn x ne

x 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 nu

dx 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

dy

dx

34 xy e 312 xdye

dxFind :

dy

dx

4 15 xy 4 14 5 ln 5xdy

dx Find :

dy

dx

Quality Practice Problems

Find :dy

dx 3lny x 2

3

13

dyx

dx x

3, 0x

x

1 1

ln 5 2

dy

dx x x

1, 0

2 ln 5x

x

5logy xFind :dy

dx

Quality Practice Problems

Find :dy

dxxy x

How 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 :dy

dxxy x

ln lnd d

y x xdx dx

ln 1dy

y xdx

ln 1xdyx x

dx

ln lny x x

ln ln xy x1 1

1 lndy

x xy dx x

Quality Practice ProblemsFind using logarithmic differentiation:

dy

dx

2

2 2

1

xxy

x

2

2 2ln ln

1

xxy

x

21ln ln 2 ln 2 ln 1

2y x x x

Differentiate: 2

1 1 1 12 ln 2 2

2 2 1

dyx

y dx x x

Quality Practice ProblemsFind using logarithmic differentiation:

dy

dx

2

1 1 1 12 ln 2 2

2 2 1

dyx

y dx x x

2

1ln 2

1

dy xy

dx x x

Substitute:

22

2 2 1ln 2

11

xx x

x 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 a1

ma

The slope of the curve:

ln 0 ln

0

a am

a a

The slope of the line:

Now, let’s set them equal…

1m

a

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 a1

ma

lny x

ln 1a

a a

ln 1a 1a e

So, our slope: 1m

e 0.368

0,0