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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).2011-03-14
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    Arturo, can you add change your comment to an answer? My answer was quite deficient...2011-03-14
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    Arturo, if indeed they take functions (rather than, say, expressions) as inputs, and their domains are disjoint, being single-variable functions in the one case, and multi-variable functions in the latter case, why are there two separate notations, when you could just define a single operator on the larger domain of single- *and* mult-variable functions?2011-03-14
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    In the multi-dimensional case $\frac{df}{dt}$ denotes the Total Derivative http://en.wikipedia.org/wiki/Total_derivative2011-03-14
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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.2011-03-15
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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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