In my experience, a problem like the following is the acid test for determining if a student really understands the Chain Rule:

Find if

The correct answer (unsimplified):

However, even students that are quite proficient with the Chain Rule can often provide the following incorrect answer:

Notice the slightly incorrect placement of the at the end of the derivative. Students can so easily get into the rhythm of just multiplying by the derivative of the inside that they can forget where the derivative of the inside should be placed.

Needless to say, a problem like this often appears on my exams as a way of separating the A students from the B students.

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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