The Chain Rule AB & BC

The chain rule answers one question: if $x$ drives $u$ and $u$ drives $y$, how fast does $y$ move per unit of $x$? The answer is the product of the two rates, $\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$. Written for a composition it is $\frac{d}{dx}f(g(x)) = f'(g(x))\cdot g'(x)$, where the outer derivative is evaluated at the inside function and the inner derivative multiplies on.

The mistakes here are structural rather than computational. The dominant one is dropping the inner factor entirely, which is silently harmless whenever the inside is $x$ and wrong everywhere else. The next is peeling a multi-layer function partway and stopping, so a triple composition loses its innermost factor. The third is misreading the structure: applying the chain rule to a product, or the product rule to a composition, because the shape was never diagnosed before a rule was chosen.