The Chain Rule was harder than I thought

Last week I taught the Chain Rule to both my Calculus I and my Business Calc class.  Let me rephrase that.  Last week I was supposed to teach the Chain Rule to both my classes.  I'm pretty sure I didn't quite get there.  At least not yet.  I was really looking forward to teaching it, because it shows up everywhere.  Also, I never remember thinking the Chain Rule was a particularly hard concept.  But maybe I'm romanticizing my beginning calculus experience.

This is my first go at teaching either class, so I'm sticking pretty close to what the books say.  I figure the authors are the experts on both the subject and the audience, so it's a good starting point.  Both books teach the Chain Rule quite differently, so I was excited to try both and compare and contrast.

Unfortunately, one thing I found across the board was that many of my students don't have a firm grasp on composition functions. Sure, they can compute fog, but ask them to go the other way--to decompose a function--and all of a sudden at least half of them look at you like you've asked them to please go swim across the Atlantic Ocean.  It was a frustrating moment as a teacher because I couldn't find a way to explain decompositions without using the typical vague words like "inside" and "outside" functions. I tried saying that the "inside" function is what has parenthesis around it, or the expression you could put parenthesis around without changing anything.  Yeah...that works for functions like

f(x)=sin(3x−5)

And

f(x)=4x2+2−−−−−−√

But when we got to

f(x)=e5x−1

they told me the inside function was e.

Not e to some power. Just e.

FAIL.

On me, not my students.

Note to self: learn how to teach the decomposition of functions.

The Chain Rule via Leibniz notation did go a bit better. We talked about how if a company that produces video games wants to know how much it is making per minute, it could take how much it makes per game sold and multiply that by how many games it sells per minute:

Similarly, if y=f(u) changes 1/2 as fast as u, and u=g(x) changes 3 times as fast as x, then we can conclude that y changes 1/2 times 3, or 1.5, times as fast as x:

y=(3x2−4)2

Or...

Another thing I tried that I stole from the Business Calc book was beginning with a "guess" for the derivative of a function such as

f(x)=(x3+5)2

For the "guess" for f'(x) we applied the Power Rule to the "inside" function and got

f′(x)=2(x3+5)

Then we found the actual derivative for f(x) by expanding it and using the Sum Rule.  We found that the derivative was the same as our guess but multiplied by 3x^2!

I thought this would be a great "AH-HA!" moment.  Alas.  It was not.

I just got a bunch of, "So, what was the guess for?"  "What's the final answer?"  "How would you enter that into MyMathLab?"

Sigh.

I think the "guess" thing really could have been powerful.  I just need to ponder how to present it better.

So, that's the Chain Rule.  Some things worked.  Some didn't.  Most didn't.  But this is one of the amazing parts of teaching mathematics--learning how others learn math.

y′=2(3x2−4)