Download Calculus 5.1 The Natural Logarithmic Function and

Calculus 5.1 The Natural Logarithmic Function and Differentiation
Definition of the Natural Logarithmic Function
The natural logarithmic function is defined by
x
1
ln x   dt , x  0
t
1
The domain of the natural logarithmic function is the set of all positive real numbers.
Properties of the Natural Logarithmic Function
The natural logarithmic function has the following properties.
1. The domain is (0,∞) and the range is (-∞,∞).
2. The function is continuous, increasing and one-to-one.
3. The graph is concave downward.
Logarithmic Properties
If a and b are positive numbers and n is rational, then the following properties are true.
1. ln(1) = 0
2. ln(ab) = lna + lnb
3. ln(an) = n ln a
a
4. ln( ) = ln a - ln b
b
Definition of e
The letter e denotes the positive real number such that
e1
ln e =  dt = 1
1 t
Derivative of the Natural Logarithmic Function
Let u be a differentiable function of x.
d
d
ln x  1 , x  0
ln u   u ' , u  0
1.
2.
dx
x
dx
u
Find the derivatives of the following functions.
a) y = ln(3x)
b) y = ln(x2 + 3x)
c) y = x ln x
d) y = ( ln x )4
Differentiate y = ln


x x2  1
3
x4  1
. Hint: rewrite first.
A Derivative Involving Absolute Value
If u is a differentiable function of x such that u ≠ 0, then
d
ln u   u '
dx
u
Find the derivative of f(x) = ln| cosx |.