Proof of Chain Rule and Intuition about what it Does I looked up a proof of the chain rule here https://web.williams.edu/Mathematics/lg5/A37W12/Chain.pdf
which made sense from a computational standpoint. However, I do not understand what intuitive details about the process of the chain rule or why it works, from this proof. 
 A: Let $h(x) = f(g(x))$. Intuitively, $h'(x)$ is a measure of how much $h$ changes per unit $x$ for a small change in $x$:
A small change $\delta x$ in $x$ causes the change $g'(x) \delta x$ in the value of g(x). The change in the value of $h(x) = f(g(x))$ as a result of the change in $g(x)$ is then given by $f'(g(x))$ multiplied by the change in $g(x)$ which was $g'(x) \delta x$. So the total change in $h$ is given by $f'(g(x))g'(x) \delta x$.
Divide both sides by $\delta x$ and take the limit of $\delta x \to 0$ (where the approximations become exact) and you have your chain rule: $h'(x) = f'(g(x)) g'(x)$.
A: If Alice can run twice as fast as Bob, then $dA/dB = 2$.  If Bob can run 3 times as fast as Carl, then $dB/dC = 3$.  How much faster can Alice run than Carl?
$$\frac{dA}{dC} = \frac{dA}{dB}\frac{dB}{dC} = 2\cdot 3 = 6,$$
as your intuition says.
A: Here is a not rigorous but I hope intuitive explanation of chain rule.
Let
$$h(x) = f(g(x))$$
then
$$\Delta h\approx f'(g(x))\Delta g$$
$$\Delta g\approx g'(x)\Delta x$$
thus
$$\Delta h\approx f'(g(x))g'(x)\Delta x\implies\frac {\Delta h}{\Delta x}\approx f'(g(x))g'(x)\implies  \frac{d h}{dx}=f'(g(x))g'(x)=\frac{df}{dg}\frac{dg}{dx}$$
For functions of several variables it works almost at the same way but with gradients and jacobians instead of the derivatives.
A: Consider two magnifiers. The first one magnifies an object $5$ times and the second one magnifies an object $3$ times. Now take an object and compose a device with  your magnifiers such that you can see the image of the image of the object through the two magnifiers. An object will be magnified $15$ times, that is the rate of change of the composite magnification is the product of rates of changes. That is what mathematicians call the chain rule.$$\frac {dw}{dx} =  \frac {dw}{du}\frac {du}{dx}$$     
