The answer to this question is yes. Students do need to know the derivatives of the inverse trig functions, at least of arcsine, arccosine, and arctangent. (The others have never shown up on a released exam question.)
Most recently, 2008 AB #26 asked students to find the tangent line to y = arctan(4x) at a given x-coordinate. I suppose students would be free to convert this equation to tan(y) = 4x and find a derivative implicitly, but the most direct way is to know the derivative of the arctangent function.
Other questions involving differentiating inverse trig functions... 1985 AB #20, 1993 BC #26, and AB #28 and BC #18 from the sample exams
Free response 2006 BC #3 also involved an arcsine, but that question was calculator active; the question could be answered using nDeriv.
You might notice a long drought between the last time a released, calculator-inactive question dealt with differentiating inverse trig functions (1993) and when we started to see them again (the sample exams were first released in 2007, I believe). If you had asked this question before the sample exam was released, I would have said that it didn't look like AP Exam was testing inverse trig derivatives anymore. But now it seems pretty clear that the test committee sees these derivatives as fair game.
Cheers, Benjamin Goldstein The Cambridge School of Weston Weston, MA
-----Original Message----- From: Tonya Lackey [mailto:firstname.lastname@example.org] Sent: Tuesday, November 30, 2010 1:01 PM To: AP Calculus Subject: [ap-calculus] Derivatives of Inverse trig functions
Do the students need to memorize the rules for derivatives of inverse trig functions for the AP exam - AB? Or will they be given? Or do they not even show up on the AB exam?