Derivatives of Exponential and Logarithmic Functions

Stewart Plus

Use the limit definition to find the derivative of ex

h

xfhxf )()(

h

xfhxfh

)()(lim

0

h

ee

h

eee

h

ee

h

xfhxf hxxhxxhx 1)()(

Find

Find

xh

h

xhx

he

h

ee

h

ee

1lim

)1(lim

00

Because 11

lim0

h

eh

hUse graphing calculator

The Derivative of ex

Therefore: The derivative of f (x) = ex is f ’(x) = ex.

Example 1

Find f’(x)

A) f(x) = 4ex – 8x2 + 7x - 14

f’(x) = 4ex – 16x + 7

B) f(x) = x7 – x5 + e3 – x + ex

f’(x) = 7x6 – 5x4 + 0 –1 + ex

= 7x6 – 5x4 –1 + ex

Example 2

Remember that e is a real number, so the power rule is used to find the derivative of xe.

Also e2 7.389 is a constant, so its derivative is 0.

Find derivatives for

A) f (x) = ex / 2 f ’(x) = ex / 2

B) f (x) = 2ex +x2

f ’(x) = 2ex + 2x

C) f (x) = -7xe – 2ex + e2 f ’(x) = -7exe-1 – 2ex

Review

xy blog ybx is equivalent to

Domain: (0, ∞)

Range: (-∞, ∞)

Range: (0, ∞)

Domain: (-∞, ∞)

xy blog

ybx * These are inverse function. The graphs are symmetric with respect to the line y=x

* There are many different bases for a logarithmic functions. Two special logarithmic functions are common logarithm (log10x or log x) and natural logarithm (logex = ln x)

Review: properties of ln

01ln

1ln

lnln

lnlnln

lnln)ln(

e

aka

bab

a

baab

k

1)

2)

3)

4)

5)

Use the limit definition to find the derivative of ln x

h

xfhxf )()(

h

xfhxfh

)()(lim

0

x

hx

hh

xhxln

1ln)ln(Find

Set s = h / x

So when h approaches 0, s also approaches o

Property 2

x

hx

hx

xln

1Multiply by 1 which is x / x

x

h

h

x

x1ln

1

Property 3

h

x

x

h

x1ln

1

s

sx

1

1ln1

s

sxs

1

1ln1

lim0

s

sx s

1

1lnlim1

0

s

sx s

1

1limln1

0e

xln

1

Definition of e Property 4: ln(e)=1x

1

Optional slide:

The Derivative of ln x

Therefore: The derivative of f (x) = ln x is f ’(x) = x

1

Example 3

Find y’ for

A)

B)

xxy ln10010 3

25 lnln eexy x

xx

xxy

10030

110030' 22

2lnln5 eexy x

01

5'

xe

xy xe

x

5

More formulas

The derivative of f(x) = bx

is f’(x) = bx ln b

The derivative of f(x) = logb x

is f’(x) =

bx ln

11

Proofs are on page 598

Example 4

Find g’(x) for

A)

B)

xxxg 10)( 10

xxxg 52 log6log)(

)10ln(1010)(' 9 xxxg

5ln

116

2ln

11)('

xxxg

5ln

6

2ln

11)('

xxg

Example 5

An Internet store sells blankets. If the price-demand

equation is p = 200(0.998)x, find the rate of change of price

with respect to demand when the demand is 400 blankets

and explain the result.

p’ = 200 (.998)x ln(0.998)

p’(400) = 200 (.998)400 ln(0.998) = -0.18.

When the demand is 400 blankets, the price is decreasing about 18 cents per blanket

Example 6A model for newspaper circulation is C(t) = 83 – 9 ln t

where C is newspaper circulation (in millions) and t is the number of

years (t=0 corresponds to 1980). Estimate the circulation and find the

rate of change of circulation in 2010 and explain the result.

t = 30 corresponds to 2010

C(30) = 83 – 9 ln30 = 52.4

C(t)’ = C’(30) =

The circulation in 2010 is about 52.4 million and is decreasing at the rate of 0.3 million per year

tt

919

3.0

30

9

Example 7: Find the equation of the tangent line to the graph

of f = 2ex + 6x at x = 0

Y = mx + b

f’(x) = 2ex + 6

m = f’(0) = 2(1) + 6 = 8

y=f (0) = 2(1) + 6(0) = 2

Y = mx + b

2 = 8(0) + b so b = 2

The equation is y = 8x + 2

Example 8: Use graphing calculator to find the points of

intersectionF(x) = (lnx)2 and g(x) = x

On your calculator, press Y=

Type in the 2 functions above for Y1 and Y2

Press ZOOM, 6:ZStandard

To have a better picture, go back to ZOOM, 2: Zoom In

*Now, to find the point of intersection (there is only 1 in this problem), press 2ND, TRACE then 5: intersect

Play with the left and right arrow to find the linking dot, when you see it, press ENTER, ENTER again, then move it to the intersection, press ENTER. From there, you should see the point of intersection

(.49486641, .49486641)