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Section 6.12 Applications of Chain Rule

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\tag{6.12.1} \end{equation}

for which it seems clear that

\begin{equation} \Partial{f}{x} = 2\tag{6.12.2} \end{equation}

But suppose we know that

\begin{equation} y=x+z\tag{6.12.3} \end{equation}

so that

\begin{equation} f = 2x+3(x+z) = 5x+3z\tag{6.12.4} \end{equation}

from which it seems equally clear that

\begin{equation} \Partial{f}{x} = 5\tag{6.12.5} \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 ,\tag{6.12.6} \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 \(F(x,y,z)=0\text{,}\) so that any of \(x,y,z\) can in principle be expressed in terms of the other two variables. Then we have

\begin{align} dz \amp= \left(\Partial{z}{x}\right)_y \,dx + \left(\Partial{z}{y}\right)_x \,dy\notag\\ \amp= \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]\notag\\ \amp= \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 .\tag{6.12.7} \end{align}

Since \(x\) and \(z\) are independent, the coefficients of \(dx\) and \(dz\) on each side of (6.12.7) must separately agree. (Equivalently, set \(x\) and \(z\) in turn equal to constants.) Thus,

\begin{align*} \left(\Partial{z}{y}\right)_x\left(\Partial{y}{z}\right)_x \amp= 1 ,\\ \left(\Partial{z}{y}\right)_x\left(\Partial{y}{x}\right)_z \left(\Partial{x}{z}\right)_y \amp= -1 . \end{align*}

The latter identity is often called the cyclic chain rule, and admits an elegant geometric interpretation. An alternative derivation is given in Section 6.13.