5.1 The Natural Logarithmic Function: Differentiation AB and BC 2015

Calculus Warm-Up 5.1 Find the derivative with respect to x:

Calculus Warm-Up 5.1 Find the indefinite integral:

Expanding Logarithmic Expressions

The Number e

Evaluating Natural Logarithmic Expressions

Derivative of ln x: The natural log function as follows: Now differentiate implicitly with respect to x: Rewrite in exponential form: Derivative of ln x

Example 1 – Differentiation of Logarithmic Functions

You Try:

Example 2 – Differentiation of Logarithmic Functions

Example 3 – You Try Find the derivative of the function:

If u > 0, If u < 0,

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 |.

Finding Relative Extrema Locate the Relative Extrema of:

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 – Solution cont’d

5.1 AB Homework Page odd, 63, 71, 77, 79, 83, 85

5.1 BC Homework Day 1: Pg odd, 63, 71, 77, 79, 83, 85, odd Day 2: MMM pgs