5.4 exponential functions: differentiation and integration
TRANSCRIPT
After this lesson, you should be able to:
•Develop properties of the natural exponential function.
•Differentiate natural exponential functions.
• Integrate natural exponential functions.
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).
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 )()( )(
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