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Chapter 8 Notes, Calculus I with Precalculus 3e Larson/Edwards
8.1
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Differentiation of Exponential Functions
The Number e
The following limits produce the same number and we call that number e.
1 x
=e
lim 1 +
x→∞
x
lim (1 + x)1/x = e
x→0
e ≈ 2.71828 . . ..
e is just a number
Since e is a number we can use it in the exponential function f (x) = ex . Why? Because it
works in many situations.
The Exponential Function
The exponential function with base e is denoted by
f (x) = ex
where x is any real number.
Properties of exponents
Let a and b be positive numbers with a 6= 1, b 6= 1 and let x and y be real numbers. Then:
A) Exponent Laws:
1. ax ay = ax+y
2. (ax )y = axy
3. (ab)x = ax bx
a x a x
4.
= x
b
b
5.
ax
= ax−y
ay
1
Chapter 8 Notes, Calculus I with Precalculus 3e Larson/Edwards
The derivative of the exponential function y = ex
If y = ex then the derivative of y is
dy
= ex
dx
If you need to use the chain rule the derivative f (u) = eu is
du
d u
[e ] = eu
dx
dx
Example 8.1.1. Solve for the following derivatives.
1. y = e2x
2. y = x2 e2x
3. y = 4e3t
4. y =
3
2 +2
ex
ex + 1
2
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Chapter 8 Notes, Calculus I with Precalculus 3e Larson/Edwards
5. y =
x
Solve this two ways.
e2x−2
6. g(t) = 2et−t
2
3
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Chapter 8 Notes, Calculus I with Precalculus 3e Larson/Edwards
8.2
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Logarithms and Derivatives of Logarithmic Functions
Let a > 0, a 6= 1 and x > 0 then
y = loga (x) ⇐⇒ x = ay
The function f (x) = loga (x) is called the logarithmic function with base a
OR
”log base a”
The natural logarithm
This is the same as before but now we use base e where e is the number we found in section
8.1. Since the log base e shows up so often we call it the natural log.
loge (x) = ln(x)
We also use log base 10 very often so we abbreviate that as
log10 (x) = log(x).
Your calculator follows the same convention.
Important Properties of Logarithms
Let b, M , N be positive real numbers with b 6= 1, and let p be any real number. Then:
1. logb (1) = 0
i.e. b0 = 1
2. logb (b) = 1
i.e. b1 = b
3. logb (bx ) = x
i.e. bx = bx
4. blogb (x) = x if x > 0
5. logb (M N ) = logb (M ) + logb (N )
6. logb
M
N
= logb (M ) − logb (N )
7. logb (M p ) = p logb (M )
8. logb (M ) = logb (N ) ⇐⇒ M = N
4
Chapter 8 Notes, Calculus I with Precalculus 3e Larson/Edwards
The derivative of the logarithmic function y = ln x
If y = ln x then the derivative of y is
dy
1
=
dx
x
If you need to use the chain rule the derivative f (u) = ln u is
du
d
[ln u] =
dx
u
Example 8.2.1. Solve for the following derivatives.
1. y = ln 5x
2. y = x2 ln(2x + 1)
3. y = 2 ln t5
4. h(x) = ln
1
2x3
5
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Chapter 8 Notes, Calculus I with Precalculus 3e Larson/Edwards
5. f (x) = ln
x+1
2x − 2
6. g(t) = ln(3t2 − 2t + 3)
6
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