Which notation should I use for the chain rule in this situation?

Question

Let $F(x,z)$ be a differentiable real-valued function that depends on $x \in \mathbb{R}$ and $z \in \mathbb{R}$, where $z=z(x,y)$, and $y \in \mathbb{R}$. Suppose that $z$ is also a differentiable function.

Is the notation of the following equation proper to describe the change in $F$ when $x$ changes, but $y$ remains constant?

\begin{align} \label{eqn1} \frac{dF}{dx}=\frac{\partial F}{\partial x} +\frac{\partial F}{\partial z}\frac{\partial z}{\partial x} \end{align}

I was tempted to use $\frac{\partial F}{\partial x}$ on the left-hand side of this equation, instead of $\frac{dF}{dx}$, to make clear that $y$ remains constant, but then I would have had $\frac{\partial F}{\partial x}$ on the right- and left-hand sides of this equation meaning different things. On the right-hand side of the equation, $\frac{\partial F}{\partial x}$ implies that $z$ is being held constant. Is there a better way to handle this situation in terms of notation?

Answer 1

If you want to really be precise with the notation, you should avoid using the same letters over and over again with different meanings, because sometimes you're using them as the names of your variables (the inputs of the functions) and sometimes, you're using it as the name of the actual function.

Here's how I'd do it. You have three functions in the game:

• First is $F:\Bbb{R}^2\to \Bbb{R}$ (where you like to call the input variables as $(x,z)$)
• Second is a function $\zeta:\Bbb{R}^2\to \Bbb{R}$
• Third, is the function $G:\Bbb{R}^2\to \Bbb{R}$ obtained via composition by $G(x,y):= F(x, \zeta(x,y))$

Now, here are two ways to write down the various chain rule applications: \begin{align} \begin{cases} \partial_1G_{(x,y)} &= \partial_1F_{(x,\zeta(x,y))} + \partial_2F_{(x,\zeta(x,y))} \cdot \partial_1\zeta_{(x,y)}\\\\ \dfrac{\partial G}{\partial x}\bigg|_{(x,y)} &= \dfrac{\partial F}{\partial x}\bigg|_{(x,\zeta(x,y))} + \dfrac{\partial F}{\partial z}\bigg|_{(x,\zeta(x,y))} \cdot \dfrac{\partial \zeta}{\partial x}\bigg|_{(x,y)} \end{cases} \end{align} I think the first is most precise, and impossible to misinterpret, because it keeps a very clear distinction between the different functions, and keeps track of where all the derivatives are evaluated. The second is still not too bad, and I think it's about as precise as you can get with Leibniz's notation.

There are some other ways I've seen people write this down, but I think it's sloppy notation (or at the very least requires a lot of interpretation), which is why I don't recommend it. Take a look at this answer, I explain how to rewrite things clearly, and also work out an example calculation in both the precise notation and the more sloppy notation.