When differentiating functions of several variables, it is essential to keep track of which variables are being held fixed. As a simple example, suppose \begin{equation} f = 2x+3y \end{equation} for which it seems clear that \begin{equation} \Partial{f}{x} = 2 \end{equation} But suppose we know that \begin{equation} y=x+z \end{equation} so that \begin{equation} f = 2x+3(x+z) = 5x+3z \end{equation} from which it seems equally clear that \begin{equation} \Partial{f}{x} = 5 \end{equation} In such cases, we adopt a more precise notation, and write \begin{equation} \left(\Partial{f}{x}\right)_y = 2 \qquad \left(\Partial{f}{x}\right)_z = 5 \end{equation} where the subscripts indicate the variable(s) being held constant.

We can now prove two useful identities about partial derivatives. Suppose that we know a relationship such as $F(x,y,z)=0$, so that any of $x,y,z$ can in principle be expressed in terms of the other two variables. Then we have \begin{eqnarray} dz &=& \left(\Partial{z}{x}\right)_y \,dx + \left(\Partial{z}{y}\right)_x \,dy \nonumber\\ &=& \left(\Partial{z}{x}\right)_y \,dx + \left(\Partial{z}{y}\right)_x \left[ \left(\Partial{y}{z}\right)_x \,dz + \left(\Partial{y}{x}\right)_z \,dx \right] \nonumber\\ &=& \left[ \left(\Partial{z}{x}\right)_y + \left(\Partial{z}{y}\right)_x\left(\Partial{y}{x}\right)_z \right]\,dx +\left(\Partial{z}{y}\right)_x\left(\Partial{y}{z}\right)_x \,dz \label{chainalg} \end{eqnarray} Since $x$ and $z$ are independent, the coefficients of $dx$ and $dz$ on each side of (\ref{chainalg}) must separately agree. (Equivalently, set $x$ and $z$ in turn equal to constants.) Thus, \begin{eqnarray} \left(\Partial{z}{y}\right)_x\left(\Partial{y}{z}\right)_x &=& 1 \\ \left(\Partial{z}{y}\right)_x\left(\Partial{y}{x}\right)_z \left(\Partial{x}{z}\right)_y &=& -1 \end{eqnarray} The latter identity is often called the cyclic chain rule, and admits an elegant geometric interpretation. (COMING SOON)
An alternative derivation is given in § {Rearranging Differentials}.