Why is $\frac{d f(g(h(x)))}{d x} = \frac{d f(g(h(x)))}{d h(x)}\frac{d h(x)}{d x}$, not $\frac{d f(g(h(x)))}{d g(h(x)))}\frac{d h(x)}{d x}$?

Question

According to the chain rule $(f(g(x))' = f'(g(x))g'(x)$.

I would have thought the latter would be correct, since I would identify $g(x)$ with $g(h(x))$ in my question title.

Answer 1

If we define a composite function as $(f\circ g \circ h)(x)$ we have two ways of interpreting it:

1. $((f \circ g)\circ h)(x)$
2. $(f \circ (g\circ h))(x)$

These are identical, and if we apply the chain rule we get:

1'. $(f\circ g)'h(x)\cdot h'(x)$

2'. $f'(g\circ h)(x)\cdot (g \circ h)'(x)$

The next step brings the two together:

$f'(g(h(x)))\cdot g'(h(x))\cdot h'(x)$