Overview

Today we conclude Topic 2.7 by developing the derivatives of e^x and ln x. Graphical presentations are included, but today’s work focuses on numerical analysis. The student activities today mirror their work from yesterday and require students to again produce a “slopes graph”: the derivative graph for these basic, and ubiquitous, functions. 

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Teaching Tips

All calculus students should have memorized at least two ordered pairs on the graph of y=e^x: (0, 1) and (1, e). If the value of e is memorized for several decimal places (see below), then the results when evaluating the derivative at x = 1 from nDeriv or Math: 8 on their calculator in Question 4 will be enlightening. We (again) ask students to plot these slope values and (again) emphasize that these slope values are the outputs of a function: the derivative function applied to y = e^x. Instead of (again) simply lecturing about a new formula to memorize, students (again) create the derivative rule. Finding a formula for the derivative of y = ln x is equally surprising to students!

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When discussing the derivative of y = ln x, our language must be precise. The relationship between “x” and “1/x” is not one of opposites or inverses. These expressions are reciprocals. Encourage students to use appropriate vocabulary in class. (Then they might actually write it on their AP Test!) Is there any algebra to review in this lesson? YES! Why is (0,∞) the domain of y = ln x ? What is the relationship between e^x and ln x? (Now we’re talking inverses.) What is the relationship between their domains and ranges?

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Exam Insights

These derivatives need to be memorized. And most students accomplish this. Knowing where the logarithm function is defined may be a scored component of an FRQ. Students should remember: domain and range matter!!