Friday, July 23, 2010

Product rule vs. trig functions

What is the best first example of the product rue?  Often, books like to use $f(x) = x e^x$.  But this is a horrible first example.  Look at the derivative: $\frac{df}{dx} = e^x + xe^x$.  Can you see the format of the product rule there?  Not in the least.  It would help to use $x^2e^x$, but the real problem is that you do not see the difference between $e^x$ and its derivative.

A much better first example might be $f(x) = x^2\sin x$.  Now $\frac{df}{dx} = 2x\sin x + x^2 \cos x$. Very nice.  You can see exactly how the product rule is used.  But there is a problem: most textbooks do not cover the derivatives of trig functions until the section after the product rule.  The reason for this appears to be the desire to keep all the trigonometric function derivatives in one place.  To get the derivative of tangent, you need the quotient rule, which should definitely be in the same section as the product rule.  What to do?

Whether textbooks like it or not, I think it is worth it to teach the derivatives of sine and cosine first, then the product and quotient rule, and then as an application of the quotient rule, do tangent, and the other three basic trig functions.  This takes a little more forethought, but the benefits clearly outweigh the costs.

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