11.2: derivatives of exponential and logarithmic functions
TRANSCRIPT
11.2: Derivatives of Exponential and Logarithmic Functions
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 –x + ex
= 7x6 – 5x4 –x + 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)