7.6: The Chain Rule and the Divide-through Rule

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If we have \(f\left(x,y\right)\) while \(x\) and \(y\) are functions of another variable, \(u\), the chain rule states that

\[\frac{df}{du}={\left(\frac{\partial f}{\partial x}\right)}_y\frac{dx}{du}+{\left(\frac{\partial f}{\partial y}\right)}_x\frac{dy}{du} \nonumber \]

If \(x\) and \(y\) are functions of variables \(u\) and \(v\); that is, \(x=x\left(u,v\right)\) and \(y=y\left(u,v\right)\), the chain rule for partial derivatives is

\[{\left(\frac{\partial f}{\partial u}\right)}_v={\left(\frac{\partial f}{\partial x}\right)}_y{\left(\frac{\partial x}{\partial u}\right)}_v+{\left(\frac{\partial f}{\partial y}\right)}_x{\left(\frac{\partial y}{\partial u}\right)}_v \nonumber \]

A useful mnemonic recognizes that these equations can be generated from the total differential by “dividing through” by \(du\). We must specify that the “new” partial derivatives are taken with \(v\) held constant. This is sometimes called the divide-through rule.

The divide-through rule is a reliable expedient for generating new relationships among partial derivatives. As a further example, dividing by \(dx\) and specifying that any other variable is to be held constant produces a valid equation. Letting w be the variable held constant, we obtain

\[\begin{align*} \left(\frac{\partial f}{\partial x}\right)_w &=\left(\frac{\partial f}{\partial x}\right)_y\left(\frac{\partial x}{\partial x}\right)_w + \left(\frac{\partial f}{\partial y}\right)_x \left(\frac{\partial y}{\partial x}\right)_w \\[4pt] &= \left(\frac{\partial f}{\partial x}\right)_y + \left(\frac{\partial f}{\partial y}\right)_x \left(\frac{\partial y}{\partial x}\right)_w \end{align*} \]

where we recognize that \({\left({\partial x}/{\partial x}\right)}_w=1\). The result is just the chain rule for\({\left({\partial f}/{\partial x}\right)}_w\) when \(f=f\left(x,y\right)\) and \(y=y\left(x,w\right)\); that is, when \(\ f=f\left(x,y\left(x,w\right)\right)\).

If we require that \(f\left(x,y\right)\) remain constant while \(x\) and \(y\) vary, we can use the divide-though rule to obtain another useful relationship from the total differential. If \(f\left(x,y\right)\) is constant, \(df\left(x,y\right)=0\). This can only be true if there is a relationship between \(x\) and \(y\). To find this relationship we use the divide-through rule to find \({\left({\partial f}/{\partial y}\right)}_f\) when \(f=f\left(x\left(y\right),y\right)\). Dividing

\[{ df=\left(\frac{\partial f}{\partial x}\right)}_ydx+{\left(\frac{\partial f}{\partial y}\right)}_xdy \nonumber \]

by \(dy\), and stipulating that \(f\) is constant, we find

\[{\left(\frac{\partial f}{\partial y}\right)}_f={\left(\frac{\partial f}{\partial x}\right)}_y{\left(\frac{\partial x}{\partial y}\right)}_f+{\left(\frac{\partial f}{\partial y}\right)}_x{\left(\frac{\partial y}{\partial y}\right)}_f \nonumber \]

Since \({\left({\partial f}/{\partial y}\right)}_f=0\) and \({\left({\partial y}/{\partial y}\right)}_f=1\), we have

\[{\left(\frac{\partial f}{\partial y}\right)}_x=-{\left(\frac{\partial f}{\partial x}\right)}_y{\left(\frac{\partial x}{\partial y}\right)}_f \nonumber \]

In Chapter 10, we find that the divide-through rule is a convenient way to generate thermodynamic relationships.