A change of variables from $\vec{r_1}$, $\vec{r_2}$ to $\vec{r}$, $\vec{R}$ is given by:

$$\vec{r} = \vec{r_1}-\vec{r_2}\text{ , }\vec{R}=c_1 \vec{r_1} + c_2 \vec{r_2}$$

I'm supposed to find $\nabla_1$ expressed in terms of $\nabla_r$ and $\nabla_R$. The suggested solution starts out with

$$\vec{R}=(X, Y, Z)\text{ , } \vec{r} = (x, y, z)$$

and then goes on to state that

$$(\nabla_1)_x = \frac{\partial}{\partial x_1} = \frac{\partial X}{\partial x_1} \frac{\partial}{\partial X} + \frac{\partial x}{\partial x_1} \frac{\partial}{\partial x} \text{ , }$$

where does (the second equality of) this last expression come from?

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It's the multivariate chain rule. –  Qiaochu Yuan Jul 17 '12 at 19:34