The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Merck Moving Beyond Passwords: FIDO Paris Seminar.pptx
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 041 handout)
1. Section 3.3
Derivatives of Exponential and
Logarithmic Functions
V63.0121.041, Calculus I
New York University
October 25, 2010
Announcements
Midterm is graded. Please see FAQ.
Quiz 3 next week on 2.6, 2.8, 3.1, 3.2
Announcements
Midterm is graded. Please
see FAQ.
Quiz 3 next week on 2.6,
2.8, 3.1, 3.2
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 2 / 34
Objectives
Know the derivatives of the
exponential functions (with
any base)
Know the derivatives of the
logarithmic functions (with
any base)
Use the technique of
logarithmic differentiation to
find derivatives of functions
involving roducts, quotients,
and/or exponentials.
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 3 / 34
Notes
Notes
Notes
1
Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
2. Outline
Recall Section 3.1–3.2
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
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 4 / 34
Conventions on power expressions
Let a be a positive real number.
If n is a positive whole number, then an
= a · a · · · · · a
n factors
a0
= 1.
For any real number r, a−r
=
1
ar
.
For any positive whole number n, a1/n
= n
√
a.
There is only one continuous function which satisfies all of the above. We
call it the exponential function with base a.
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 5 / 34
Properties of exponential Functions
Theorem
If a > 0 and a = 1, then f (x) = ax
is a continuous function with domain
(−∞, ∞) and range (0, ∞). In particular, ax
> 0 for all x. For any real
numbers x and y, and positive numbers a and b we have
ax+y
= ax
ay
ax−y
=
ax
ay
(negative exponents mean reciprocals)
(ax
)y
= axy
(fractional exponents mean roots)
(ab)x
= ax
bx
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 6 / 34
Notes
Notes
Notes
2
Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
3. Graphs of various exponential functions
x
y
y = 1x
y = 2xy = 3x
y = 10x
y = 1.5x
y = (1/2)xy = (1/3)x
y = (1/10)x
y = (2/3)x
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 7 / 34
The magic number
Definition
e = lim
n→∞
1 +
1
n
n
= lim
h→0+
(1 + h)1/h
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 8 / 34
Existence of e
See Appendix B
We can experimentally verify
that this number exists and
is
e ≈ 2.718281828459045 . . .
e is irrational
e is transcendental
n 1 +
1
n
n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
1000 2.71692
106
2.71828
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 9 / 34
Notes
Notes
Notes
3
Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
4. Logarithms
Definition
The base a logarithm loga x is the inverse of the function ax
y = loga x ⇐⇒ x = ay
The natural logarithm ln x is the inverse of ex
. So
y = ln x ⇐⇒ x = ey
.
Facts
(i) loga(x1 · x2) = loga x1 + loga x2
(ii) loga
x1
x2
= loga x1 − loga x2
(iii) loga(xr
) = r loga x
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 10 / 34
Graphs of logarithmic functions
x
y
y = 2x
y = log2 x
(0, 1)
(1, 0)
y = 3x
y = log3 x
y = 10x
y = log10 x
y = ex
y = ln x
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 11 / 34
Change of base formula for logarithms
Fact
If a > 0 and a = 1, and the same for b, then
loga x =
logb x
logb a
Proof.
If y = loga x, then x = ay
So logb x = logb(ay
) = y logb a
Therefore
y = loga x =
logb x
logb a
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 12 / 34
Notes
Notes
Notes
4
Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
5. Upshot of changing base
The point of the change of base formula
loga x =
logb x
logb a
=
1
logb a
· logb x = (constant) · logb x
is that all the logarithmic functions are multiples of each other. So just
pick one and call it your favorite.
Engineers like the common logarithm log = log10
Computer scientists like the binary logarithm lg = log2
Mathematicians like natural logarithm ln = loge
Naturally, we will follow the mathematicians. Just don’t pronounce it
“lawn.”
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 13 / 34
Outline
Recall Section 3.1–3.2
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
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 14 / 34
Derivatives of Exponential Functions
Fact
If f (x) = ax
, then f (x) = f (0)ax
.
Proof.
Follow your nose:
f (x) = lim
h→0
f (x + h) − f (x)
h
= lim
h→0
ax+h − ax
h
= lim
h→0
ax ah − ax
h
= ax
· lim
h→0
ah − 1
h
= ax
· f (0).
To reiterate: the derivative of an exponential function is a constant times
that function. Much different from polynomials!
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 15 / 34
Notes
Notes
Notes
5
Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
6. The funny limit in the case of e
Remember the definition of e:
e = lim
n→∞
1 +
1
n
n
= lim
h→0
(1 + h)1/h
Question
What is lim
h→0
eh − 1
h
?
Answer
If h is small enough, e ≈ (1 + h)1/h
. So
eh − 1
h
≈
(1 + h)1/h h
− 1
h
=
(1 + h) − 1
h
=
h
h
= 1
So in the limit we get equality: lim
h→0
eh − 1
h
= 1
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 16 / 34
Derivative of the natural exponential function
From
d
dx
ax
= lim
h→0
ah − 1
h
ax
and lim
h→0
eh − 1
h
= 1
we get:
Theorem
d
dx
ex
= ex
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 17 / 34
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
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 18 / 34
Notes
Notes
Notes
6
Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
7. Examples
Examples
Find derivatives of these functions:
e3x
ex2
x2
ex
Solution
d
dx
e3x
= 3e3x
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 19 / 34
Outline
Recall Section 3.1–3.2
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
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 20 / 34
Derivative of the natural logarithm function
Let y = ln x. Then
x = ey
so
ey dy
dx
= 1
=⇒
dy
dx
=
1
ey
=
1
x
We have discovered:
Fact
d
dx
ln x =
1
x
x
y
ln x
1
x
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 21 / 34
Notes
Notes
Notes
7
Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
8. The Tower of Powers
y y
x3
3x2
x2
2x1
x1
1x0
x0
0
ln x x−1
x−1
−1x−2
x−2
−2x−3
The derivative of a power
function is a power function
of one lower power
Each power function is the
derivative of another power
function, except x−1
ln x fills in this gap precisely.
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 22 / 34
Outline
Recall Section 3.1–3.2
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
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 23 / 34
Other logarithms
Example
Use implicit differentiation to find
d
dx
ax
.
Solution
Let y = ax
, so
ln y = ln ax
= x ln a
Differentiate implicitly:
1
y
dy
dx
= ln a =⇒
dy
dx
= (ln a)y = (ln a)ax
Before we showed y = y (0)y, so now we know that
ln a = lim
h→0
ah − 1
h
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 24 / 34
Notes
Notes
Notes
8
Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
9. Other logarithms
Example
Find
d
dx
loga x.
Solution
Let y = loga x, so ay
= x. Now differentiate implicitly:
(ln a)ay dy
dx
= 1 =⇒
dy
dx
=
1
ay ln a
=
1
x ln a
Another way to see this is to take the natural logarithm:
ay
= x =⇒ y ln a = ln x =⇒ y =
ln x
ln a
So
dy
dx
=
1
ln a
1
x
.
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 25 / 34
More examples
Example
Find
d
dx
log2(x2
+ 1)
Answer
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 26 / 34
Outline
Recall Section 3.1–3.2
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
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 27 / 34
Notes
Notes
Notes
9
Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
10. A nasty derivative
Example
Let y =
(x2 + 1)
√
x + 3
x − 1
. Find y .
Solution
We use the quotient rule, and the product rule in the numerator:
y =
(x − 1) 2x
√
x + 3 + (x2 + 1)1
2(x + 3)−1/2 − (x2 + 1)
√
x + 3(1)
(x − 1)2
=
2x
√
x + 3
(x − 1)
+
(x2 + 1)
2
√
x + 3(x − 1)
−
(x2 + 1)
√
x + 3
(x − 1)2
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 28 / 34
Another way
y =
(x2 + 1)
√
x + 3
x − 1
ln y = ln(x2
+ 1) +
1
2
ln(x + 3) − ln(x − 1)
1
y
dy
dx
=
2x
x2 + 1
+
1
2(x + 3)
−
1
x − 1
So
dy
dx
=
2x
x2 + 1
+
1
2(x + 3)
−
1
x − 1
y
=
2x
x2 + 1
+
1
2(x + 3)
−
1
x − 1
(x2 + 1)
√
x + 3
x − 1
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 29 / 34
Compare and contrast
Using the product, quotient, and power rules:
y =
2x
√
x + 3
(x − 1)
+
(x2 + 1)
2
√
x + 3(x − 1)
−
(x2 + 1)
√
x + 3
(x − 1)2
Using logarithmic differentiation:
y =
2x
x2 + 1
+
1
2(x + 3)
−
1
x − 1
(x2 + 1)
√
x + 3
x − 1
Are these the same?
Which do you like better?
What kinds of expressions are well-suited for logarithmic
differentiation?
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 30 / 34
Notes
Notes
Notes
10
Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
11. Derivatives of powers
Question
Let y = xx
. Which of these is true?
(A) Since y is a power function,
y = x · xx−1
= xx
.
(B) Since y is an exponential
function, y = (ln x) · xx
(C) Neither
x
y
1
1
Answer
(A) This can’t be y because xx
> 0 for all x > 0, and this function decreases at
some places
(B) This can’t be y because (ln x)xx
= 0 when x = 1, and this function does not
have a horizontal tangent at x = 1.
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 31 / 34
Derivatives of power functions with any exponent
Fact (The power rule)
Let y = xr
. Then y = rxr−1
.
Proof.
y = xr
=⇒ ln y = r ln x
Now differentiate:
1
y
dy
dx
=
r
x
=⇒
dy
dx
= r
y
x
= rxr−1
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 33 / 34
Summary
Derivatives of logarithmic and exponential functions:
y y
ex
ex
ax
(ln a) · ax
ln x
1
x
loga x
1
ln a
·
1
x
Logarithmic Differentiation can allow us to avoid the product and
quotient rules.
V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 34 / 34
Notes
Notes
Notes
11
Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010