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Let function be $f(x)=u\cdot v$ where $u$ and $v$ are in terms of $x$. Then how to make someone understand that $f'(x) = uv' + u'v $ only using chain rule?

My attempt: I don't even think it is possible but I may be wrong.

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    See here: https://en.m.wikipedia.org/wiki/Chain_rule#Proofs – 2017-01-30
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    @Rohan He's asking to prove product rule with chain rule. – 2017-01-30

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We have the multivariate chain rule:

$$\frac d{dx}f(u,v)=\frac{\partial f}{\partial u}\frac{du}{dx}+\frac{\partial f}{\partial v}\frac{dv}{dx}$$

and with $f(u,v)=u\cdot v$, we get

$$\frac d{dx}u\cdot v=vu'+uv'$$

since

$$\frac\partial{\partial u}u\cdot v=v$$

$$\frac\partial{\partial v}u\cdot v=u$$

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    Brilliant! I never thought about this, but it's quite natural. – 2017-01-30
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    @CameronWilliams well, there was [that one question](http://math.stackexchange.com/questions/2055481/why-not-learn-the-multi-variate-chain-rule-in-calculus-i) about how the multivariate chain rule can do pretty much everything... – 2017-01-30
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    What a coincidence! ;) – 2017-01-30