Using chain rule to differentiate $f(x)=a(x)b(x)$? Why can I not apply the chain rule to a product in the following way.
If we have some product:
$$f(x)=a(x)b(x)$$
Consider the multiplication of b by a as another’s function so that:
$$f(b(x))=ab$$
So that 
$\frac{df}{dx} = f’(b)b’(x)$
Something feels very wrong. But I can’t put my finger on it.

You can't use the chain rule because this is not a composition. Just calling it one does not make it one. For example, suppose $a(x)=x+1$ and $b(x)=x^2$. Then your first line says
$$
f(x) = x^2(x+1) = x^3+ x^2
$$
so
$$
f(b(x)) = f(x^2) = (x^2)^3 + (x^2)^2
$$
so not the same as "$ab$".
