## Section6.12Applications 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.