# Honing intuition is hard work (but worth it)

I thought I’d kill birds with stones and (1) try using LaTeX equations while (2) explaining how I honed my calculus classes intuition. Granted, the idea is simple and I think most teachers teach it this way, but I didn’t and my students got confused. So on Day 2 of the chain rule, I had them come in and do the following right away:

Find the derivatives:

A1: 
B1: 
C1: 

A2: 
B2: 
C2: 

A3: 
B3: 
C3: 

Bonus Problems:

D1: Find the derivative of .

D2: Find the derivative of .

And in fact, they were very successful. Instead of showing them a big equation and how to break it down, I instead started small and built it up. And showed them each part of what was going on. And by the end, my students were pretty capable chain-rule-appliers.

1. It is important to first introduce  as  because it makes the composition of functions easier to see. (Plus, really surprisingly and disappointingly, some of my students didn’t realize they were the same thing!)

2. There is something really exhilerating about writing really long answers to problems. They love it. I love it. It makes us feel important and like we’re doing something extraordinarily complex. Which we are (to some extent).

3. I think it’s important to polish off this whole chain rule business with something that shows the students that all these things that they’re doing works. Physically worsk. And are just like what we’ve been doing. So what I did after we worked on this worksheet is we went back to a derivative we had initially solved with the chain rule: . I asked them what the equation of the tangent line to this function was at a particular value of . And then I showed them that what we found for the tangent line worked graphically. They saw the tangent line hit the curve in front of them. At this point, there weren’t many gasps (they knew it was going to work), but I think it drove home that (a) hell yeah, it works! and i’m. not. lying. to. you. and (b) this is no different than everything else we’ve been doing. (We’ve been finding tangent lines to tons of functions. These functions are just longer and more gross looking.)