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## dy/dx = 6tan(x)cos(3x) Correct?

A recent advertisement for the software vendor WebAssign caught my eye. Let' see what you think...

The problem posed was: Find the derivative of y = 2cos(3x-pi) with respect to x.

The answer written by the student was "dy/dx = 6 sin(3x)." Is this right or wrong? The expected answer probably is "dy/dx = -6sin(3x-pi)."

The intent of the ad was to show that WebAssign, in comparsion to its competitors, was able to score the given answer as correct. The claim: it is able to recognize unexpected but correct answers to free response questions...."like a living, breathing professor."

So, what's my quandary? First, a professor would probably expect the student to show his/her work, and thus the given answer would not be a surprise or counted as wrong.

And second, an experienced professor might be suspicious of a the missing minus sign and the absence of the (-pi) term. And, only the student's work would resolve the doubts.

For example, did the student know that d(cos x)/dx = -sin(x)? Or did he/she make a common mistake and use d(cos x)/dx = sin(x)? And, though able to do the chain rule in part to get the 3 as a multiplier, did the student just eliminate the term (-pi) because the derivative of a constant is 0?

But, you say I am too suspicious? And I say, no...not since I have students who do these types of things.

In fact, given the student's answer, I would expect that a double error had occurred...rather than assume that the student did a horizontal shift of some sort, either first noting that 2cos(3x-pi) = -2cos(3x) and taking the derivative...or taking the derivative and then noticing that -6sin(3x-pi) = 6sin(3x).

To me, both of these possibilities are unlikely (as the student's work would bear out). Thus, it is not enough for an online homework system to just look at final answers...which is why professors will always be preferred (and needed)!

Finally, is any correct answer correct? For example, would you give full credit if the student wrote dy/dx = 6tan(x)cos(3x)?