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

Let $h(x)=f(g(x))$
$$\begin{align}h'(x) &= \lim_{t\to0}\frac{h(x+t)-h(t)}{t}\\&=\lim_{t\to0}\frac{f(g(x+t))-f(g(x))}{t}\\\end{align}$$
Now there are two possible cases,

*

*$g(x+t)=g(x)$
In this case, $h(x+t)=h(x)$, and $h'(x)=0$, and $h'(x)=f'(g(x))\cdot g'(x)=0$
is satisfied.


*$g(x+t)\to g(x)$
In this case we can write the limit as,
$$\lim_{t\to0}\frac{f(g(x+t))-f(g(x))}{t} = \frac{f(g(x+t))-f(g(x))}{g(x+t)-g(x)}\cdot \frac{g(x+t)-g(x)}t = f'(g(x))\cdot g'(x)$$
We do not consider the case where $\lim_{t\to0} g(x+t) \not \to g(x)$ since continutity and differentiability are requisite conditions here.
