Derivatives of exponential and logarithmic functions
1. Section 3.3
Derivatives of Exponential and
Logarithmic Functions
V63.0121, Calculus I
March 10/11, 2009
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2. Outline
Derivative of the natural exponential function
Exponential Growth
Derivative of the natural logarithm function
Derivatives of other exponentials and logarithms
Other exponentials
Other logarithms
Logarithmic Differentiation
The power rule for irrational powers
. . . . . .
4. Derivatives of Exponential Functions
Fact
If f(x) = ax , then fβ² (x) = fβ² (0)ax .
Proof.
Follow your nose:
f(x + h) β f(x) ax+h β ax
fβ² (x) = lim = lim
h h
hβ0 hβ0
ax ah β ax ah β 1
= ax Β· fβ² (0).
= ax Β· lim
= lim
h h
hβ0 hβ0
. . . . . .
5. Derivatives of Exponential Functions
Fact
If f(x) = ax , then fβ² (x) = fβ² (0)ax .
Proof.
Follow your nose:
f(x + h) β f(x) ax+h β ax
fβ² (x) = lim = lim
h h
hβ0 hβ0
ax ah β ax ah β 1
= ax Β· fβ² (0).
= ax Β· lim
= lim
h h
hβ0 hβ0
To reiterate: the derivative of an exponential function is a constant
times that function. Much different from polynomials!
. . . . . .
6. The funny limit in the case of e
Remember the deο¬nition of e:
( )
1n
= lim (1 + h)1/h
e = lim 1 +
n
nββ hβ0
Question
eh β 1
What is lim ?
h
hβ0
. . . . . .
7. The funny limit in the case of e
Remember the deο¬nition of e:
( )
1n
= lim (1 + h)1/h
e = lim 1 +
n
nββ hβ0
Question
eh β 1
What is lim ?
h
hβ0
Answer
If h is small enough, e β (1 + h)1/h . So
[ ]h
(1 + h)1/h β 1
eh β 1 (1 + h) β 1 h
β = = =1
h h h h
. . . . . .
8. The funny limit in the case of e
Remember the deο¬nition of e:
( )
1n
= lim (1 + h)1/h
e = lim 1 +
n
nββ hβ0
Question
eh β 1
What is lim ?
h
hβ0
Answer
If h is small enough, e β (1 + h)1/h . So
[ ]h
(1 + h)1/h β 1
eh β 1 (1 + h) β 1 h
β = = =1
h h h h
eh β 1
=1
So in the limit we get equality: lim
h
hβ0
. . . . . .
9. Derivative of the natural exponential function
From
( )
ah β 1 eh β 1
dx
ax
a= =1
lim and lim
dx h h
hβ0 hβ0
we get:
Theorem
dx
e = ex
dx
. . . . . .
10. Exponential Growth
Commonly misused term to say something grows exponentially
It means the rate of change (derivative) is proportional to the
current value
Examples: Natural population growth, compounded interest,
social networks
. . . . . .
11. Examples
Examples
Find these derivatives:
e3x
2
ex
x2 ex
. . . . . .
12. Examples
Examples
Find these derivatives:
e3x
2
ex
x2 ex
Solution
d 3x
e = 3ex
dx
. . . . . .
13. Examples
Examples
Find these derivatives:
e3x
2
ex
x2 ex
Solution
d 3x
e = 3ex
dx
d x2 2d 2
e = ex (x2 ) = 2xex
dx dx
. . . . . .
14. Examples
Examples
Find these derivatives:
e3x
2
ex
x2 ex
Solution
d 3x
e = 3ex
dx
d x2 2d 2
e = ex (x2 ) = 2xex
dx dx
d 2x
x e = 2xex + x2 ex
dx
. . . . . .
15. Outline
Derivative of the natural exponential function
Exponential Growth
Derivative of the natural logarithm function
Derivatives of other exponentials and logarithms
Other exponentials
Other logarithms
Logarithmic Differentiation
The power rule for irrational powers
. . . . . .
16. Derivative of the natural logarithm function
Let y = ln x. Then
x = ey so
. . . . . .
17. Derivative of the natural logarithm function
Let y = ln x. Then
x = ey so
dy
ey =1
dx
. . . . . .
18. Derivative of the natural logarithm function
Let y = ln x. Then
x = ey so
dy
ey =1
dx
dy 1 1
=β = y=
dx e x
. . . . . .
19. Derivative of the natural logarithm function
Let y = ln x. Then
x = ey so
dy
ey=1
dx
dy 1 1
=β = y=
dx e x
So:
Fact
d 1
ln x =
dx x
. . . . . .
20. Derivative of the natural logarithm function
y
.
Let y = ln x. Then
x = ey so
dy
ey=1
dx l
.n x
dy 1 1
=β = y=
dx e x
. x
.
So:
Fact
d 1
ln x =
dx x
. . . . . .
21. Derivative of the natural logarithm function
y
.
Let y = ln x. Then
x = ey so
dy
ey=1
dx l
.n x
dy 1 1 1
=β = y= .
dx e x x
. x
.
So:
Fact
d 1
ln x =
dx x
. . . . . .
22. The Tower of Powers
yβ²
y
The derivative of a power
x3 3x2 function is a power
function of one lower
x2 2x1
power
x1 1x0
x0 0
? ?
xβ1 β1xβ2
xβ2 β2xβ3
. . . . . .
23. The Tower of Powers
yβ²
y
The derivative of a power
x3 3x2 function is a power
function of one lower
x2 2x1
power
x1 1x0 Each power function is
the derivative of another
0
x 0
power function, except
xβ1 xβ1
?
xβ1 β1xβ2
xβ2 β2xβ3
. . . . . .
24. The Tower of Powers
yβ²
y
The derivative of a power
x3 3x2 function is a power
function of one lower
x2 2x1
power
x1 1x0 Each power function is
the derivative of another
0
x 0
power function, except
xβ1 xβ1
ln x
xβ1 β1xβ2 ln x ο¬lls in this gap
precisely.
xβ2 β2xβ3
. . . . . .
25. Outline
Derivative of the natural exponential function
Exponential Growth
Derivative of the natural logarithm function
Derivatives of other exponentials and logarithms
Other exponentials
Other logarithms
Logarithmic Differentiation
The power rule for irrational powers
. . . . . .
26. Other logarithms
Example
dx
Use implicit differentiation to ο¬nd a.
dx
. . . . . .
27. Other logarithms
Example
dx
Use implicit differentiation to ο¬nd a.
dx
Solution
Let y = ax , so
ln y = ln ax = x ln a
. . . . . .
28. Other logarithms
Example
dx
Use implicit differentiation to ο¬nd a.
dx
Solution
Let y = ax , so
ln y = ln ax = x ln a
Differentiate implicitly:
1 dy dy
= (ln a)y = (ln a)ax
= ln a =β
y dx dx
. . . . . .
29. Other logarithms
Example
dx
Use implicit differentiation to ο¬nd a.
dx
Solution
Let y = ax , so
ln y = ln ax = x ln a
Differentiate implicitly:
1 dy dy
= (ln a)y = (ln a)ax
= ln a =β
y dx dx
Before we showed yβ² = yβ² (0)y, so now we know that
2h β 1 3h β 1
β 0.693 β 1.10
ln 2 = lim ln 3 = lim
h h
hβ0 hβ0
. . . . . .
31. Other logarithms
Example
d
Find log x.
dx a
Solution
Let y = loga x, so ay = x.
. . . . . .
32. Other logarithms
Example
d
Find log x.
dx a
Solution
Let y = loga x, so ay = x. Now differentiate implicitly:
dy dy 1 1
(ln a)ay = 1 =β =y =
dx dx a ln a x ln a
. . . . . .
33. Other logarithms
Example
d
Find log x.
dx a
Solution
Let y = loga x, so ay = x. Now differentiate implicitly:
dy dy 1 1
(ln a)ay = 1 =β =y =
dx dx a ln a x ln a
Another way to see this is to take the natural logarithm:
ln x
ay = x =β y ln a = ln x =β y =
ln a
dy 11
=
So .
dx ln a x
. . . . . .
34. More examples
Example
d
log (x2 + 1)
Find
dx 2
. . . . . .
35. More examples
Example
d
log (x2 + 1)
Find
dx 2
Answer
dy 1 1 2x
= (2x) =
2+1 (ln 2)(x2 + 1)
dx ln 2 x
. . . . . .
36. Outline
Derivative of the natural exponential function
Exponential Growth
Derivative of the natural logarithm function
Derivatives of other exponentials and logarithms
Other exponentials
Other logarithms
Logarithmic Differentiation
The power rule for irrational powers
. . . . . .
37. A nasty derivative
Example β
(x2 + 1) x + 3
. Find yβ² .
Let y =
xβ1
. . . . . .
38. A nasty derivative
Example β
(x2 + 1) x + 3
. Find yβ² .
Let y =
xβ1
Solution
We use the quotient rule, and the product rule in the numerator:
[β ] β
(x β 1) 2x x + 3 + (x2 + 1) 1 (x + 3)β1/2 β (x2 + 1) x + 3(1)
β² 2
y=
(x β 1)2
β β
(x2 + 1) (x2 + 1) x + 3
2x x + 3
+β β
=
(x β 1) (x β 1)2
2 x + 3(x β 1)
. . . . . .
39. Another way
β
(x2 + 1) x + 3
y=
xβ1
1
ln y = ln(x + 1) + ln(x + 3) β ln(x β 1)
2
2
1 dy 2x 1 1
β
=2 +
x + 1 2(x + 3) x β 1
y dx
So
( )
dy 2x 1 1
β
= + y
x2 + 1 2(x + 3) x β 1
dx
β
( )
(x2 + 1) x + 3
2x 1 1
β
= +
x2 + 1 2(x + 3) x β 1 xβ1
. . . . . .
40. Compare and contrast
Using the product, quotient, and power rules:
β β
(x2 + 1) (x2 + 1) x + 3
2x x + 3
β²
+β β
y=
(x β 1) (x β 1)2
2 x + 3(x β 1)
Using logarithmic differentiation:
β
( )2
(x + 1) x + 3
2x 1 1
β²
β
y= +
x2 + 1 2(x + 3) x β 1 xβ1
. . . . . .
41. Compare and contrast
Using the product, quotient, and power rules:
β β
(x2 + 1) (x2 + 1) x + 3
2x x + 3
β²
+β β
y=
(x β 1) (x β 1)2
2 x + 3(x β 1)
Using logarithmic differentiation:
β
( )2
(x + 1) x + 3
2x 1 1
β²
β
y= +
x2 + 1 2(x + 3) x β 1 xβ1
Are these the same?
. . . . . .
42. Compare and contrast
Using the product, quotient, and power rules:
β β
(x2 + 1) (x2 + 1) x + 3
2x x + 3
β²
+β β
y=
(x β 1) (x β 1)2
2 x + 3(x β 1)
Using logarithmic differentiation:
β
( )2
(x + 1) x + 3
2x 1 1
β²
β
y= +
x2 + 1 2(x + 3) x β 1 xβ1
Are these the same?
Which do you like better?
. . . . . .
43. Compare and contrast
Using the product, quotient, and power rules:
β β
(x2 + 1) (x2 + 1) x + 3
2x x + 3
β²
+β β
y=
(x β 1) (x β 1)2
2 x + 3(x β 1)
Using logarithmic differentiation:
β
( )2
(x + 1) x + 3
2x 1 1
β²
β
y= +
x2 + 1 2(x + 3) x β 1 xβ1
Are these the same?
Which do you like better?
What kinds of expressions are well-suited for logarithmic
differentiation?
. . . . . .
44. Derivatives of powers
Let y = xx . Which of these is true?
(A) Since y is a power function, yβ² = x Β· xxβ1 = xx .
(B) Since y is an exponential function, yβ² = (ln x) Β· xx
(C) Neither
. . . . . .
45. Derivatives of powers
Let y = xx . Which of these is true?
(A) Since y is a power function, yβ² = x Β· xxβ1 = xx .
(B) Since y is an exponential function, yβ² = (ln x) Β· xx
(C) Neither
. . . . . .
46. Itβs neither! Or both?
If y = xx , then
ln y = x ln x
1 dy 1
= x Β· + ln x = 1 + ln x
y dx x
dy
= xx + (ln x)xx
dx
Each of these terms is one of the wrong answers!
. . . . . .
47. Derivative of arbitrary powers
Fact (The power rule)
Let y = xr . Then yβ² = rxrβ1 .
. . . . . .
48. Derivative of arbitrary powers
Fact (The power rule)
Let y = xr . Then yβ² = rxrβ1 .
Proof.
y = xr =β ln y = r ln x
Now differentiate:
1 dy r
=
y dx x
dy y
= r = rxrβ1
=β
dx x
. . . . . .