In calculus, does $\frac{dy}{dx}$ represents the slope of a function? If so, what does $d$ in the numerator represents in $\frac{d}{dx}$? And why does $\frac{d}{dx} y=\frac{dy}{dx}$?
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1The notation comes from Liebniz, I believe, who thought in terms of "infinitesimally small change in y" in proportion to "infinitesimally small change in x". See: https://en.wikipedia.org/wiki/Leibniz%27s_notation – Elizabeth S. Q. Goodman Apr 04 '17 at 08:17
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It is a single symbol, and not a fraction. It just behaves like one some times. Specifically, it doesn't have a numerator, and there is no $d$. There is only $\frac d{dx}$. – Arthur Apr 04 '17 at 08:20
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@ElizabethS.Q.Goodman: Leibniz, not Liebniz. – celtschk Apr 04 '17 at 08:25
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I know, but it's too late to fix typos. – Elizabeth S. Q. Goodman Apr 04 '17 at 08:40
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http://math.stackexchange.com/questions/1551124/notation-for-higher-degree-derivatives/2000010#2000010 – Apr 04 '17 at 22:27
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The notation comes from Leibniz, back in that day they dealt primarily with infinitesimal numbers, such as $\epsilon$. Back then it wasn't rigorously defined as it didn't happen until the 19th and 20th century that mathematics got a rigorous foundation.
His idea was that if $y=f(x)$ then the derivative is the infinitesimal change in $y$ value divided by the infinitesimal change in $x$ value. This can be put into a proper fraction and used as a normal fraction with all the rules still apply. In the 60s I think it was the concept of infinitesimal was rigorously defined and now can be shown to be true and rigorous.
From this the notation follows quite naturally and all the identities we use.
celtschk
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Zelos Malum
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In modern mathematics and hyperreals, $\epsilon$ is usually denoted for an infinitesimal and $\omega$ is commonly a transfinite number there. – Zelos Malum Apr 04 '17 at 08:28