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When I studied derivative, I sometimes saw notation $\frac{d}{dx}2x=2$ and sometimes $\frac{\partial}{\partial x}2x=2$. What is the definition and difference between those notations?

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Usually $\partial$ is reserved for functions with more than 1 argument. –  wj32 Nov 2 '12 at 9:39

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$\frac{\text d}{\text dx} $ is the total derivative $\qquad\qquad\frac{\partial}{\partial x} $ is the partial derivative

For a function of one variable say $f(x)$$${\text d f }={\partial f}\iff\frac{\partial f}{\partial x}=\frac{\text df}{\text dx}$$

For a function of two variables say $g(x,y)$ $${\text d g }=\frac{\partial g}{\partial x}\text dx+\frac{\partial g}{\partial y}\text dy\iff\frac{\text dg}{\text dx}=\frac{\partial g}{\partial x}+\frac{\partial g}{\partial y}\frac{\text dy}{\text dx}$$

For a function of three variables say $h(x,y,z)$ $${\text d h }=\frac{\partial h}{\partial x}\text dx+\frac{\partial h}{\partial y}\text dy+\frac{\partial h}{\partial z}\text dz\iff\frac{\text dh}{\text dx}=\frac{\partial h}{\partial x}+\frac{\partial h}{\partial y}\frac{\text dy}{\text dx}+\frac{\partial h}{\partial z}\frac{\text dz}{\text dx}$$

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If $f: \mathbb{R} \rightarrow \mathbb{R}$, then $f$ of course is of one variable and the derivative is with respect to this one variable, say $x$. Therefore, we use the notation $\frac{df}{dx}$. however, if $f$ is a multivariable function, that is $f: \mathbb{R}^n \rightarrow \mathbb{R}$, So $f(x_1,...,x_n) = w$, then we have to take derivatives with respect to one of the $x_i$'s leaving the other $x$'s as constants, and thats why we use the notation $$\frac{\partial f}{\partial x_i}$$

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I think you're stretching the meaning of the word “why” here. It's just a convention, but a useful one; though the usefulness of this convention doesn't really becomes apparent until you consider functions of the form $f(x,y)$ in which $y$ is sometimes considered to be a function of $x$ (or situatations analogous to this one). –  Harald Hanche-Olsen Nov 2 '12 at 9:49

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