Calculate derivatives of inverse trig functions
I can derive the derivative formulas for arcsin x, arccos x, and arctan x using right triangle trig
I can calculate derivatives of inverse trig functions, including those that require the chain rule
Quick Lesson Plan
Getting Triggy With It (It’s unlikely your students were around in 1993, but they certainly may know the Will Smith song or even the Phish version) walks students through many important algebraic and trigonometric concepts: inverse functions, trig identities, right triangle properties, and more. Most students will be able to recall the relationships needed to complete the entire activity. Additionally, problems 2 and 3 repeat the concepts and procedures from problem 1, allowing students to use their previous work for assistance. This is very much a student-guided discovery of the derivatives for the arcsin(x), arccos(x), and arctan(x) functions.
The CED requires students to know the derivatives of six inverse trigonometric functions. Derivatives for arcsin(u), arccos(u), arctan(u), and arccot(u), where u is a function of x, are likely to appear later when students encounter antidifferentiation, so these forms should be emphasized over the other three. Once we demonstrate to students that the derivatives of inverse co-functions are opposites, they can more easily remember all six forms. And although the derivatives of arcsec(u) and arccsc(u) rarely appear as integrands, it can be valuable to present for dramatic effect. The AP Calculus CED does not explicitly list which of the inverse trig functions will appear on future AP Tests. Still, we are hard-pressed to find MCQs or FRQs that utilize the arcsec(u) or arccsc(u) derivatives.
These derivatives may appear in either the MCQ section of the test or be embedded in an FRQ.
Misapplications of the chain rule and challenges with manipulating square root notation are the most common issues associated with this lesson. Students will not be asked to derive these formulas; memorization is the most efficient method of dealing with these derivatives.