# Chain rule notation for composite functions

Suppose I have a function $f(x, y, g(x, y))$

How would I express $\frac{\partial f}{\partial x}$? Using the chain rule, you'd naturally come up with $\frac{\partial f}{\partial x} + \frac{\partial f}{\partial g} \frac{\partial g}{\partial x}$, except in this expression, $\frac{\partial f}{\partial x}$ is really only the partial derivative of $f$ with respect to that one parameter, and not $x$. So, my question is, what notation would I use to show this differentiation that is less ambiguous and meaningless than $\frac{\partial f}{\partial x} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial g} \frac{\partial g}{\partial x}$?

## 1 Answer

You have a function $f(x,y,z)$ presumably, and then you take a composition $h(x,y) = f(x,y,g(x,y))$. The chain rule here is

$$\frac{\partial h}{\partial x} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial z}\frac{\partial g}{\partial x}$$

and similarly for $y$. You should verify that on your own, and check a couple examples to convince yourself.

• I don't think you understood what I was trying to ask, but your answer explained what I wanted to know anyway. My error was in thinking that the $\partial f / \partial x$ on the left side which held only $y$ fixed (which you wrote as $\partial h / \partial x$ to remove ambiguity) represented the same thing as the $\partial f / \partial x$ on the right side, which held both $y$ and $g(x, y)$ fixed. – glaba Dec 17 '14 at 5:36
• Introducing the extra letters h and z made it clear what exactly was being held constant. – glaba Dec 17 '14 at 5:38