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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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"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). – Arturo Magidin Mar 14 '11 at 19:59
Arturo, can you add change your comment to an answer? My answer was quite deficient... – JavaMan Mar 14 '11 at 20:49
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In the multi-dimensional case $\frac{df}{dt}$ denotes the Total Derivative en.wikipedia.org/wiki/Total_derivative – Brian Mar 14 '11 at 21:48
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@Quine42: Mainly, because you want to distinguish between functions which depend on a single variable and those that do not; the partial derivative is a generalization of the regular derivative, and does not carry as much information as the regular derivative (e.g., a function of two variables may have partials wrt both variables at a point, but not be continuous at the point; with derivatives, that is impossible). You can abuse notation and use $\partial$ for single variable functions, but that implies that the function is "really" a multi-variable function. – Arturo Magidin Mar 15 '11 at 3:31
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@Quine42: Also, see the link Brian gives; the regular $d$ notation for multivariable functions has a different meaning. – Arturo Magidin Mar 15 '11 at 3:34
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