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$$\frac{\partial f}{\partial x}(x+y)=\frac{\partial }{\partial x}f(x+y)$$

I was just wondering what the left-hand side mean. (or how to do the operation based on the notation of the LHS, given a specific function $f$)

In addition, when is such commutation true? Under what conditions?

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    I think this is a definition of the LHS. They are telling you that the partial derivative of $f(x + y)$ with respect to $x$ is denoted by what is written on the LHS. Observe that $x + y$ is a function of two variables, while, I presume, $f$ is a function of one variable. So $f(x + y)$ becomes a function of two variables, and you're just taking the partial derivative of this function with respect to $x$; that is, $(x,y)\mapsto f(x + y)$.2012-02-05
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    My initial reaction after a little bit of thought is that the LHS is bad notation.2013-02-20

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