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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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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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    $\begingroup$ Brilliant! I never thought about this, but it's quite natural. $\endgroup$ Jan 30, 2017 at 16:29
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    $\begingroup$ @CameronWilliams well, there was that one question about how the multivariate chain rule can do pretty much everything... $\endgroup$ Jan 30, 2017 at 16:41
  • $\begingroup$ What a coincidence! ;) $\endgroup$ Jan 30, 2017 at 16:59

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