TITLE: Intuition behind chain rule
QUESTION [15 upvotes]: What is the intuition behind chain rule in mathematics in particular why there is a multiplication in between?

REPLY [3 votes]: In terms of differentials, we know that if the variables $x,y,z$ are related by $y = f(x)$ and $z = g(y)$, then

$dy = f'(x) dx$
$dz = g'(y) dy$

If differentials are even the slightest bit reasonable to think about, then we should be able to substitute, and get

$dz = g'(y) f'(x) dx$

or, if you prefer,

$dz = g'(f(x)) f'(x) dx$