And a student (one of my top students, I might add) suggested we "divide out an ln."
Let's ignore the fact that dividing by any number other than 1 would change the expression.
Dividing by ln? So...somehow there's not a connection that ln is meaningless without an argument. "Dividing by ln" is akin to "dividing by √ " or "dividing by cos." An empty square root or an empty cosine doesn't have any kind of value, and really doesn't mean a thing.
I was further disturbed when I gave my Business Calculus students a function like
and was told that in order to find the derivative, we should use the Product Rule.
Do students view ln as some sort of constant? Like e? It seems maybe so if the function above is thought of as a product and if we can indeed "divide out ln."
I decided we needed to revisit the Chain Rule.
I started by showing a slide that had a composite function at the top and four expressions beneath it such as:
I asked my students to tell me why we needed to use the Chain Rule and then to identify the derivative of the outside function (holding the inside) and the derivative of the inside function. The next slide highlighted the former in red and the latter in blue:
I think I will start with this type of presentation the next time I teach the Chain Rule. Giving the students a limited amount of options to start out with seemed to worked fairly nicely.
There are still definitely some issues. But I now have a better understanding of what needs to be emphasized in terms of composite functions. I will try to make them more densely populated in my algebra and pre-calculus classes from now on.
If you're interested, here's the slideshow we worked through:
Test 3 Review