1. The Natural Logarithmic Function
Section 5.1
The Natural Logarithmic Function

2. THE NATURAL LOGARITHMIC FUNCTION
Definition: The natural logarithmic function is the function defined by
ln(x) = 1 𝑥 1 𝑡 dt x>0
Remember this from the
graphing activity

3. THE DERIVATIVE OF THE NATURAL LOGARITHMIC FUNCTION
From the Fundamental Theorem of Calculus, Part 1, we see that
𝑑ln(x) 𝑑𝑥 = 1 𝑥 x>0
Remember we discussed
this in class

4. PROPERTIES OF THE NATURAL LOGARITHMIC FUNCTION
Using calculus, we can describe
the natural logarithmic function.
Remember x>0
1. ln(x) is an increasing function, since
𝑑ln(x) 𝑑𝑥 = 1 𝑥 >0
2. The graph of ln(x) is concave downwards, since
𝑑 2 ln(x) 𝑑 𝑥 2 = −1 𝑥 2 <0

5. lim 𝑥→0 ln 𝑥 =−∞ lim 𝑥→∞ ln 𝑥 =∞
This is consistent with what we know about the graph of ln(x)
lim 𝑥→∞ ln 𝑥 =∞

6. THE DERIVATIVE OF THE NATURAL LOGARITHM AND THE CHAIN RULE
𝑑𝑙𝑛 𝑔 𝑥 𝑑𝑥 = 1 𝑔(𝑥) 𝑔′(𝑥)

7. ANTIDERIVATIVES INVOLVING THE NATURAL LOGARITHM
Theorem:
Remember the domain of the natural log is positive real numbers.
𝑑ln 𝑥 𝑑𝑥 = 1 𝑥
1 𝑥 𝑑𝑥=𝑙𝑛 𝑥 +c
HW: p. 329#43-60 x 3 by Tue
p. 330#63-87 x 3 by Wed

8. LOGARITHMIC DIFFERENTIATION
How can we use this information to help us solve problems?
Take logarithms of both sides of an equation y = f (x) and use the laws of logarithms to simplify.
Differentiate implicitly with respect to x.
Solve the resulting equation for y ′.

9. Example: Differentiate y=ln(3x2-2)3
Rewrite: y=3ln(3x2-2)
𝑦 ′ =3 𝑑𝑙𝑛(3 𝑥 2 −2) 𝑑𝑥
𝑦 ′ = 1 𝑔(𝑥) 𝑔′(𝑥)
𝑦 ′ =3 1 (3 𝑥 2 −2) (6x)
𝑦 ′ = 18𝑥 (3 𝑥 2 −2)

10. ANTIDERIVATIVES OF SOME TRIGONOMETRIC FUNCTIONS
Memorize these
tan 𝑥 𝑑𝑥=𝑙𝑛 sec⁡(𝑥) +𝐶
cot 𝑥 𝑑𝑥=𝑙𝑛 sin⁡(𝑥) +𝐶
sec 𝑥 𝑑𝑥=𝑙𝑛 sec 𝑥 tan⁡(𝑥) +𝐶
csc 𝑥 𝑑𝑥=𝑙𝑛 csc 𝑥 cot⁡(𝑥) +𝐶