# Notation of derivative

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

$\frac{\text d}{\text dx}$ 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)$$${\text d h }=\frac{\partial h}{\partial x}\text dx+\frac{\partial h}{\partial y}\text dy+\frac{\partial h}{\partial z}\text dz\iff\frac{\text dh}{\text dx}=\frac{\partial h}{\partial x}+\frac{\partial h}{\partial y}\frac{\text dy}{\text dx}+\frac{\partial h}{\partial z}\frac{\text dz}{\text dx}$$ - 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 $$\frac{\partial f}{\partial x_i}$$ - 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