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Why do we have dy/dx with the regular d, and 'del y/del x' with the 'funny' d? I can easily find definitions for each expresion, but the definitions appear to be logically equivalent. However, they are informal enough that it is possible that I am not understanding the definitions properly.

Specifically, can anyone show me specific inputs for which the d/dx and del/delx operators return different outputs? And do they return functions, or values?

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    @Quine42: Also, see the link Brian gives; the regular $d$ notation for multivariable functions has a different meaning.2011-03-15

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As Arturo Magidin puts it:

"Regular $d$" are derivatives of a single-variable function relative to that single variable. $\partial$ means "partial derivative", it refers to a function of several variables, when we take derivatives relative to only one of the variables, treating the others as constant. They are meant to be applied to different animals ($\frac d{dx}$ to single-variable functions of $x$, $\frac\partial{\partial x}$ to multi-variable functions).