Homework Homework Assignment #18 Read Section 3.10 Page 191, Exercises: 1 – 37 (EOO) Quiz next time Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page Find the inverse g(x) of with domain x ≥ 0 and calculate g′(x) in two ways, using Theorem 1 and by direct calculation. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 191 Use Theorem 1 to calculate g′(x) where g(x) is the inverse of f (x). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page Let g(x) be the inverse of f (x) = x 3 +2x +4. Calculate g(7) and then calculate g′(7). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 191 Calculate g(b) and g′(b) where g is the inverse of f. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page Let f (x) = x n and g(x) = x 1/n. Compute g′(x) using Theorem 1 and check your answer using the Power Rule. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 191 Compute the derivative at the point without using a calculator. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 191 Find the derivative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 191 Find the derivative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 191 Find the derivative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 191 Find the derivative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Chapter 3: Differentiation Section 3.10: Derivatives of General Exponential and Logarithmic Functions Jon Rogawski Calculus, ET First Edition

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In this section, we consider the derivatives of various exponential and logarithmic functions. First, theorem 1 gives the derivative of a base raised to a power that is a function of x.

Example, Page 197 Find the derivative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Natural logarithms are frequently used in physics and engineering, and their derivative is found using Theorem 2. Remember that we can only find logarithms of positive numbers.

Example, Page 197 Find the derivative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 197 Find the derivative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Two of the most important calculus facts about exponential functions are given in the following box.

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Applying the Chain Rule to the derivative of the natural log, we have the following equality.

Example, Page 197 Find the derivative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Recall from Math Analysis the change of base formula for logarithms. From the second equation, if b > 0 and b ≠ 1, then the derivative of log b x is given by:

Example, Page 197 Compute the derivative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Find the derivatives of the functions in Figure 1.

Example, Page 197 Compute the derivative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 197 Evaluate the derivative using logarithmic differentiation. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework Homework Assignment #19 Read Section 3.11 Page 197, Exercises: 1 – 49 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company