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Q: Is there an intuitive proof for the chain rule?

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Physicist: The chain rule is a tool from calculus that says that if you have one function “nested” inside of another, , then the derivative of the whole mess is given by . When you multiply the argument by some amount, the graph of the function gets squished by the same amount. The slope (derivative) is “rise over run” and the run just became half as long, so the slope just got twice as big. Chain rule:It’s worth pointing out that, like all calc rules, it doesn’t matter that this rule only talks about two functions. If you have something like , then you can treat as one function, and you’ll find that after running through the chain rule once you’ll be faced with another, simpler, chain rule problem: