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Apologies if this question has been answered, but I haven't been able to find it.

Is there a way to "derive" the product rule for two scalar functions as a dot product

$\dfrac{d}{dx} \left[fg\right] \ = \ \langle f, f' \rangle \cdot \langle g', g \rangle$

or is this simply a notational device with nothing special about it?

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As a different dot product, yes.

$h(t) = f(t)g(t)$ is the composite of the maps $F(t) = (f(t),g(t))$ and $m(x,y)=xy$. Moreover $dm(x,y) = (y,x)$.

By the (multivariable) chain rule we therefore have $h'(t) = dm(F(t)) \cdot F'(t) = (g(t),f(t)) \cdot (f'(t),g'(t))$.

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  • $\begingroup$ Thank you. Could you explain how $dm(x,y) = (y,x)$? $\endgroup$ – Abbas Jaffary Oct 5 '14 at 17:43
  • $\begingroup$ $dm(x,y) = (\partial m/\partial x, \partial m/\partial y)$. Perhaps you know this as the gradient of $m$, $\nabla m$. $\endgroup$ – user180040 Oct 5 '14 at 18:06
  • $\begingroup$ Ah yes, thank you! Should have caught that. $\endgroup$ – Abbas Jaffary Oct 5 '14 at 18:17

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