Calculate derivatives of products of differentiable functions
Use identities to rewrite tangent, cotangent, secant, and cosecant functions and then apply derivative rules to find formulas for their derivatives
Use the rules for derivatives of trigonometric functions in association with other derivative rules
I can develop trig derivatives by using identities and other derivative formulas
I can apply a variety of derivative rules to complex differentiable functions
Quick Lesson Plan
These are the last of the six trig derivatives to be memorized. The context for this lesson is straightforward, but a valuable review of the trig identities for tan x, cot x, sec x, and csc x. This lesson provides repeated applications of the quotient rule and trig identities are needed to simplify the final derivative formulas.
Many of our students are skilled at memorizing (think AP Bio, AP Chem, or AP Psych) and most will have a technique for remembering these derivatives. Be sure to point out, however, that any student unable (or unwilling!) to memorize these formulas can always take the time to derive them using the methods of today’s activity. Most will see the light and have the formulas memorized quickly. Knowing that derivatives of the “co” functions will produce a negative sign (which we pronounce “opposite” --- especially when reciting the quadratic formula) helps students better remember the derivatives. Referring often to a graphical representation of the derivative (for example, a graph of the tangent function shows only positive slopes; a graph of the cotangent function shows only negative slopes) can help students remember the formulas, too.
These derivatives must be memorized as they are tested individually and as intermediate steps in more complex questions. Found on both the MC and FRQ sections of the test, students will be successful on these questions with consistent exposure to derivatives of trig functions.
Failure of memory is the biggest obstacle: forgetting trig identities, forgetting which derivatives require a negative sign, or forgetting to include the angle when writing a trig expression!!!