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In physics, it seems like the use of $\dfrac{dy}{dx}$ and $\dfrac{\partial y}{\partial x}$ are used somewhat interchangeably.

My understanding is that, technically $\dfrac{dy}{dx}$ is only appropriate where $y$ only has a single variable (x), but may have many other constants. By contrast $\dfrac{\partial y}{\partial x}$ says $y$ has more than just a single $x$ variable, but for now we're holding those variables as constants and treating the $x$ as the only variable.

Essentially, $\dfrac{dy}{dx}$ and $\dfrac{\partial y}{\partial x}$ should give you the same result when applied to $y=x^2+3x-2\theta+\lambda^2 $, except $\dfrac{dy}{dx}$ implies that $\theta$ and $\lambda$ are constants and always will be, while $\dfrac{\partial y}{\partial x}$ suggests that at least one of $\lambda$ or $\theta$ is a variable.

Is this understanding correct, or have I missed something?

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    $\begingroup$ Pretty much accurate; there's no real difference except for the connotations. $\endgroup$ – Tim Ratigan Dec 23 '13 at 16:30
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I think this article answers your question http://www.math.dartmouth.edu/archive/m23f04/public_html/PartialDifferentiation.pdf

Taking an ordinary derivative of a function of one or more variables, it is assumed that all the variables depend on x.

For partial derivatives it is assumed that all the variables are independent.

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  • $\begingroup$ Great source - thanks $\endgroup$ – Alex Dec 23 '13 at 17:28

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