Composite function derivatives

Question

Is this statment right in derivative of compoaite three functions $(f \circ g \circ h)'(x) = (f\circ g)' (h(x))$

Answer 1

I don't think so. At least not in general case:

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

so, it follows $(f \circ g)'(h(x)) = f'(g(h(x)))\ g'(h(x))$. It's quite similar to (1) but not the same. It would be the same, for example, when $h'(x) = 1$.

Answer 2

No, it's not right.

Set $F=f \circ g$. Then, by the usual chain rule, $$ (f \circ g \circ h)'(x) = (F \circ h)'(x) = F'(h(x)) \, h'(x) = (f \circ g)'(h(x)) \, h'(x). $$