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I still don't fully understand why differentials can sometimes be treated as fractions and sometimes cannot. Here's something I tried playing around—using the quotient rule to evaluate $\frac{dy}{dx}$

$$ \frac{d}{dx} \left( \frac{dy}{dx}\right) = \frac{\frac{d^2 y}{dx}dx - dy\frac{d^2x}{dx}}{dx^2} = \frac{d^2y}{dx^2} - \frac{dy\,\, d^2 x}{dx^3}$$

Could someone explain this?

More specifically, if I understand correctly, one should be able to find the differential $$d \left( \frac{dy}{dx}\right) = \frac{d^2y}{dx^2}dx - \frac{d^2 x}{dx^2}dy$$ I'm not quite sure what this means.

user3697389
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  • "$\frac{\mathrm{d}^2 y}{\mathrm{d}x}$" and "$\frac{\mathrm{d}^2 x}{\mathrm{d}x}$" are typos?... Or perhaps: What process do these particular collections of symbols describe? – Eric Towers Dec 17 '17 at 05:09
  • If you really wanted to go that way, you could perhaps "rationalize" that $d^2 x = 0$ since the second differential of a linear must be null. However, any such line of thought is fraught with many pitfalls. I would suggest perusing Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio? for a start. – dxiv Dec 17 '17 at 05:11
  • Point taken. My follow up question would be, what, then, does the differential notation mean for second derivatives ($d^2 x$)? I thought differentials were supposed to represent a derivative without regard to differentiating variable but clearly $d^2 x$ stands for 0 in regards to $x$ but can have some meaning as an infinitisemal otherwise? – user3697389 Dec 17 '17 at 06:03

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