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

The best way to think about the derivative is: if $f$ is differentiable at $x$, then
\begin{equation*}
f(x + \Delta x) \approx f(x) + f'(x) \Delta x.
\end{equation*}
The approximation is good when $\Delta x$ is small.  This is practically the definition of $f'(x)$.
Now suppose $f(x) = g(h(x))$, and $h$ is differentiable at $x$, and $g$ is differentiable at $h(x)$.  Then
\begin{align*}
f(x + \Delta x) & = g(h(x+\Delta x)) \\
&\approx g(h(x) + h'(x) \Delta x) \\
&\approx g(h(x)) + g'(h(x)) h'(x) \Delta x.
\end{align*}
Comparing this with the equation above suggests that
\begin{align*}
f'(x) = g'(h(x)) h'(x).
\end{align*}
Many other rules about derivatives can be derived easily in this way.
