5.1 The Natural Logarithmic Function: Differentiation

AB and BC2015

Calculus Warm-Up 5.1

3 2 25 4y y y x

Find the derivative with respect to x:

Calculus Warm-Up 5.1

1dxx

Find the indefinite integral:

Expanding Logarithmic Expressions

22

3 2

10ln

9

ln 3x 2

6xln

5

x 3ln

x x 1

ln10 ln 9

The Number e

Derivative of ln x: 0x

lny xThe natural log function as follows:

ye x

Now differentiate implicitlywith respect to x: 1ye y

Rewrite in exponential form:

1ln

dx

dx x

1 1y

ye x

Derivative of ln x

Example 1 – Differentiation of Logarithmic Functions

Example 1 – Differentiation of Logarithmic Functions

You Try:

e. ln 1d

xdx

1

2 2x

Example 2 – Differentiation of Logarithmic Functions

2 2

3

( 1)Differentiate ( ) ln

2 1

x xf x

x

1/ 22 2 3ln ( 1) ln 2 1x x x

2 31ln 2ln( 1) ln 2 1

2x x x

2

2 3

1 2 1 6'( ) 2

1 2 2 1

x xf x

x x x

2

2 3

1 4 3

1 2 1

x x

x x x

Be sure you see the benefit of applying logarithmic properties

before you differentiate.

Example 3 – You Try

31

ln1

xf x

x

Find the derivative of the function:

2

2

3 1f x

x

If u > 0,

, ln ln .d d u u

u u and u udx dx u u

If u < 0,

, ln .d u

u u and udx u

The natural log function is undefined for negative numbers

Example 4 – Derivative Involving Absolute Value

You Try: Find the derivative of

f(x) = ln | cosx |.

Solution:

Example 5 – Derivative Involving Absolute Value

You Try: Find the derivative of

f(x) = ln | secx+tanx |.

secy x

Finding Relative Extrema

• Locate the Relative Extrema of:

2y ln x 2x 3

Sometimes, it is convenient to use logarithms as aids in differentiating nonlogarithmic functions.

This procedure is called logarithmic differentiation.

Logarithmic Differentiation (Optional)

Example 6 – Logarithmic Differentiation

Find the derivative of

Solution:Note that y > 0 for all x ≠ 2. So, ln y is defined. Begin by taking the natural logarithm of each side of the equation.

Then apply logarithmic properties and differentiate implicitly.

Finally, solve for y'.

Example 6 – Solutioncont’d

5.1 AB Homework

• Page 330 45-59 odd, 63, 71, 77, 79, 83, 85

5.1 BC Homework

• Day 1: Pg. 330 45-59 odd, 63, 71, 77, 79, 83, 85, 93-97 odd

• Day 2: MMM pgs.191-192