I'm halfway through my first year of teaching AP Calculus to high school seniors. It's been going generally well, but I'm feeling like I really could have done better getting them into the Chain Rule.
I started with it the same basic way that I did with the Product and Quotient Rules -- showing that the rule worked for elementary polynomials and could save us some calculation time. But, in retrospect, the Chain Rule is such a fundamental part of much of the rest of differentiation that I feel like there could have been more that would help them understand how it works and how the concept ties together.
Is the u-substitution notion a good idea? Our class is generally much more comfortable with the $f'(x)$ notation, and as a result I stayed away from the $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$ format. Instead, I did a lot of hand-waving around the "inside function" and the "outside function" that hasn't taken hold in all my students as well as I might have hoped.
Any suggestions about what works in your calculus classrooms?