Intuition behind chain rule What is the intuition behind chain rule in mathematics in particular why there is a multiplication in between?

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


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*$dy = f'(x) dx$

*$dz = g'(y) dy$


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


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


or, if you prefer,


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

