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Did Euler have juxtaposition of $dx$ to $f'(x)$ to denote multiplication of a "very small quantity" to $f'(x)$ to obtain another "very small quantity" $dy$? This seems to imply that $\frac{dy}{dx}$ is a fraction, but did Euler ever think so? If not I guess he saw differential notation as notation shortening helpful in, for example, integral notation.

This question differs from others asked here because I'm curious about Euler, and I'm also aware of nonstandard analysis and how it is often taught today that $dy/dx$ is not a fraction.

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Yes, to Euler $dy/dx$ is literally a fraction, and furthermore $dx$ and $dy$ are both literally zero. From Euler, Foundations of Differential Calculus, 1755, translated by J. D. Blanton, Springer, 2000:

To anyone who asks what an infinitely small quantity in mathematics is, we can respond that it really is equal to $0$. There is really not such a great mystery lurking in this idea as some commonly think and thus have rendered the calculus of the infinitely small suspect to so many. (§83)

If we accept the notation used in the analysis of the infinite, then $dx$ indicates a quantity that is infinitely small, so that both $dx = 0$ and $a dx = 0$, where $a$ is any finite quantity. Despite this, the geometric ratio $a dx : dx$ is finite, namely $a : 1$. For this reason these two infinitely small quantities $dx$ and $adx$, both being equal to $0$, cannot be confused when we consider their ratio. In a similar way, we will deal with infinitely small quantities $dx$ and $dy$. Although these are both equal to $0$, still their ratio is not that of equals. Indeed, the whole force of differential calculus is concerned with the investigation of the ratios of any two infinitely small quantities of this kind. (§86)

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Viktor Blasjo
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  • No wonder he (supposedly) made the mistake involving uniform continuity where all infinitesimals are equal to zero but some are more zero then others... – user21820 Jul 06 '15 at 08:15