In the above proof, Spivak suddenly shifts to $f'(a)$ notation which has confused me about whats really going on. How do we apply chain (2-2) rule which was derived for the derivative and not partial derivative? What are the intermediate steps?
From what I figure $$D(f \circ g)(a^j)=Df(a)\circ Dh(a^j) $$ How does the composition on the right-hand side become a product?