CAS Derivatives

Here’s a fun problem from my calculus class today, enhanced by CAS. As a set-up, our last unit focused on interpreting the meaning of derivatives with multiple interpretations of the definition of the derivative as the only algebraic work they’ve done.  In that unit, the students discovered that vertical translations on functions didn’t change their derivatives, and horizontal translations on functions changed the corresponding derivatives by the same horizontal translation.  From their work with derivatives of power functions using a definition of the derivative, they hypothesized and proved \frac{d}{dx}(x^n)=n*x^{n-1} for natural (and a few other) values of n.  Knowing nothing else, I posed this.

Use your CAS, determine the derivatives of y=ln(x), y=ln(2x), y=ln(3x), and y=ln(4x).  Use your results to hypothesize the derivative of y=ln(n*x).  Justify your claim.

The following image from the first part shows that the pattern is easy to spot.

Unfortunately, I posed the problem with only 10 minutes remaining in class, but the students clearly knew \frac{d}{dx}(ln(n*x))=\frac{1}{x}, but the looming question wasWHY.  With a couple minutes to spare, one guessed rules of logarithms might apply, but not having used them since their first semester exam, he didn’t recall the property.  Some colleagues may argue that I should have insisted on my students having those rules memorized in advance, but I firmly believe that this particular problem actually gave a reason for my students to relearn their logarithm properties.

I let the awkward moment hang there until another called out with glee,
ln(n*x)=ln(n)+ln(x)“, to which a third exclaimed, “and ln(n) is a constant, making the derivative of ln(n*x) the same as the derivative of
ln(x),” clearly using her understanding of the effect of translations on functions and their derivatives that she learned in the last unit.

Two other nice ideas emerged:

  1. They thought it convenient that \frac{d}{dx}(ln(n*x))=\frac{1}{x}, but now really want to know why.
  2. A few observant ones noticed that ln(x) and \frac{1}{x} have different domains.  To these, I pointed out the warning at the bottom of the CAS screen above that most had completely overlooked when getting their initial answers.

These questions still linger for the class, but I argue that the use of CAS in my calculus class this afternoon

  • granted everyone equal access to the same mathematics,
  • emphasized the importance of logarithm properties in a more meaningful way than any pleas from me could have done,
  • left a need in many to discover why \frac{d}{dx}(ln(n*x))=\frac{1}{x}, and
  • raised domain issues that ultimately will lead to a deeper understanding for the existence of the absolute value in \int{\frac{1}{x}dx}=ln(|x|).

It was a good end to our first day back after Spring Break and left my students wanting to know more.  What a great place to leave a group of students, thanks to CAS.


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