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5.4 Exponential Functions: Differentiation
and Integration
After this lesson, you should be able to:
•Develop properties of the natural exponential function.
•Differentiate natural exponential functions.
• Integrate natural exponential functions.
Definition of the Natural Exponential Function and Figure 5.19
The Natural Exponential Functions
Theorem 5.10 Operations with Exponential Functions
Properties of the Natural Exponential Function
Let’s consider the derivative of the natural exponential function.
Difficult WayGoing back to our limit definition of the derivative:
h
eee
dx
d xhx
h
x
0lim
First rewrite the exponential using exponent rules.
h
eee xhx
h
0lim
Next, factor out ex. h
ee hx
h
1lim
0
Since ex does not contain h, wecan move it outside the limit.
h
xfhxfxf
dx
dh
0lim
h
ee
h
h
x 1lim
0
Derivatives of Natural Exponential Functions
Substituting h = 0 in the limit expression results in the
indeterminate form , thus we will need to determine it.
0
0
We can look at the graph of and observe what
happens as x gets close to 0. We can also create a table of values close to either side of 0 and see what number we are closing in on.
x
exf
x 1)(
x -.1 -.01 -.001 .001 .01 .1f(x) .95 .995 .999 1.0005 1.005 1.05
Graph
At x = 0, f(0) appears to be 1.
Table
As x approaches 0, f(x) approaches 1.
We can safely say that from the last slide that 11
lim0
h
eh
h
Thus
Rule 1: Derivative of the Natural Exponential Function
xxh
h
xx eeh
eee
dx
d
1
1lim
0
xx eedx
d
The derivative of the natural exponential function is itself.
Easy WayGoing back to Differentiation of Inverse Function :
Let , then . We have already known thatxey yx ln
ydy
dx 1
xx eydx
dye
dx
d][
So
Example 1 Find the derivative of f(x) = x2ex .
Solution Do you remember the product rule? You will need it here.
xeexxf
exxfxx
x
2)(
)(2
2
Product Rule:(1st)(derivative of 2nd) + (2nd)(derivative of 1st)
)2()( xxexf x Factor out the common factor xex.
Example 2 Find the derivative of 2
3
)2()( tetf
Solution We will need the chain rule for this one.
tt
t
eetf
etf
2
1
2
3
)2(2
3)('
)2()(
Chain Rule:(derivative of the outside)(derivative of the inside)
Why don’t you try one: Find the derivative of . 2x
exf
x
To find the solution you should use the quotient rule. Choose from the expressions below which is the correct use of the quotient rule.
x
ex'f
x
2
4
2 2
x
xeexx'f
xx
4
22
x
exxex'f
xx
No that’s not the right choice.
Remember the Quotient Rule:
(bottom)(derivative of top) – (top)(derivative of bottom)
(bottom)²
Try again. Return
Good work!
The quotient rule results in . 4
xx2
x
2xeexx'f
Now simplify the derivative by factoring the numerator and canceling.
3
x
4
x
4
xx2
x
2xex'f
x
2xxe
x
2xeexx'f
What if the exponent on e is a function of x and not just x?
Rule 2: If f(x) is a differentiable function then
)()()( xfeedx
d xfxf
In words: the derivative of e to the f(x) is an exact copy of e to the f(x) times the derivative of f(x).
Theorem 5.11 Derivative of the Natural Exponential Function
Example 3 Find the derivative of xexf 3)(
Solution We will have to use Rule 2. The exponent, 3x is a function of x whose derivative is 3.
3)(
)(3
3
x
x
exf
exf
An exact copy ofthe exponential function Times the derivative of
the exponent
Example 4 Find the derivative of 12 2
)( xexf
Solution
)4()(
)(12
12
2
2
xexf
exfx
x
12 2
4)( xxexf
Again, we used Rule 2. So the derivative is the exponential function times the derivative of the exponent.
Or rewritten:
Example 5 Differentiate the function tt
t
ee
etf
)(
2)(
)()()(
tt
tttttt
ee
eeeeeetf
SolutionUsing the quotient rule
2
0202
)()(
tt
tt
ee
eeeetf
Keep in mind that thederivative of e-t is e-
t(-1) or -e-t
Recall that e0 = 1.
2)(
2)(
tt eetf
Distribute et into the ( )’s
You try: Find the derivative of . xexf 5
Click on the button for the correct answer.
x
ex'f
x
52
5 5
xex'f x 55
No, the other answer was correct.
Remember when you are doing the derivative of e raised to the power f(x) the solution is e raised to the same power times thederivative of the exponent.
What is the derivative of ?
Try again. Return
5x
Good work!!
Here is the derivative in detail.
x
ex'f
xex'f
xex'f
xdx
dex'f
x
x
x
x
-
52
5
52
5
552
1
5
5
5
5
5
2
1
Example 6 A quantity growing according to the law where Q0 and k are positive constants and t
belongs to the interval experiences exponential growth.Show that the rate of growth Q’(t) is directly proportional to the amount of the quantity present.
kteQtQ 0)(
,0
Solution
)()(
)(
0
0
tkQkeQtQ
eQtQkt
kt
Remember: To say Q’(t) is directly proportional to Q(t) means that for some constant k, Q’(t) = kQ(t) which was easy to show.
Example 7 Find the inflection points of 2
)( xexf
Solution We must use the 2nd derivative to find inflection points.
2
2
2
1
2
1
12
02
1220
24)(
2)]2([2)(
2)(
)(
2
2
2
2
2
2
22
22
2
2
x
x
x
e
xe
eexxf
exexxf
xexf
exf
x
x
xx
xx
x
x
First derivative
Product rule for second derivative
Simplify
Set equal to 0.
Exponentials never equal 0.
Set the other factor = 0.
Solve by square root of both sides.
To show that they are inflection points we put them on a number line and do a test with the 2nd derivative:
707.02
2
707.0
2
2
Intervals Test Points Value
,
,
,
22
22
22
22 -1
0
1
f ”(-1)= 4e-1 – 2e-1 =2e-1 > 0
f ”(0)=0 – 2 = –2 < 0
f ”(1)= 4e-1 – 2e-1 = 2e-1 > 0
22
2
24)(
)(2 xx
x
eexxf
exf
+ - +
Since there is a sign change across the potential inflection points,
2
1
,2
2eand are inflection points.
2
1
,2
2e
In this lesson you learned two new rules of differentiation and used rules you have previously learned to find derivatives of exponential functions.
The two rules you learned are:Rule 1: Derivative of the Natural Exponential Function
xx eedx
d
Rule 2: If f(x) is a differentiable function then
)()()( xfeedx
d xfxf
Integrals of Natural Exponential Functions
Each rules of differentiation has a corresponding integration rule.
xx eedx
d
Rule 2 : If f(x) is a differentiable function then
)()()( xfeedx
d xfxf
Rule 1 : Derivative of the Natural Exponential Function
Rule 2 : If f(x) is a differentiable function then
Rule 1 : Integral of the Natural Exponential Function Cedxe xx
Cexdfe xfxf )()( )(
Theorem 5.12 Integration Rules for Exponential Functions
Example 8 Find
Solution We must use Rule 2 of Integration.
Make an f(x) or u in the “d ”
Apply the Rule 2 of Integration
dxe x 254
Ce
xdedxe
x
xx
25
2525
5
4
)25(5
44
Example 9 Find
Solution We must use Rule 2 of Integration.
Make an f(x) or u in the “d ”
Apply the Rule 2 of Integration
dxxe x 223
Ce
xdedxxe
x
xx
2
22
2
222
4
3
)2(4
33
Example 10 Find
Solution We must use Rule 2 of Integration.
Try to make an f(x) or u in the “d ”
Apply the Rule 2 of Integration
dxex
x 2/13
5
Ce
xde
xdedxex
x
x
xx
2
2
22
/1
2/1
2/1/13
2
5
)/1(2
5
)/1(2
55
Made an f(x) or u in the “d ”
Example 11 Find
Solution We will use Integration Rule of Basic Trig of Functions.
Use Rule 2 of Derivative of N. Exp. Func.
dxexe xx )tan(22
Ce
ede
xdee
dxexe
x
xx
xx
xx
|cos|ln
)()tan(2
1
)()tan(2
1
)tan(
2
22
22
22
2
Made an u in the “d ”
Apply the Integration Rule for Basic Trig Function
Example 12 Find dxe
ex
x
1
0
Solution We must use Rule 2 of Integration.
1
1
0
1
0
1
0
1
0
1
0
1ln
)ln(
)(1
)(1
)(
e
e
ede
ede
xde
edx
e
e
x
xx
xx
x
x
x
x
Use Rule 2 of Derivative of N. Exp. Func.
Apply the Log Rule for Integration
Homework
5.4 P. 356 Q xxxxxxxxxxxx
