# Differentiation of a function with respect to a product of variables

How would you preceed to differentiate a function with respect to a product of variables were the product does not appear together in the function. For example:

If $y=r\sin(\theta)$. How would you proceed with: $\frac{\mathrm{d} y}{\mathrm{d} (r\theta)}$

Thanks for your help in advance.

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## 1 Answer

This expression is not well-defined. $y$ is a function of two variables. If you want one of them to be $r \theta$, you have to specify what the other one is to know what to keep constant.

As a simpler example, suppose $y = \theta$. If I want my two variables to be $u = r \theta$ and $v = r$, then $y = \frac{u}{v}$ and $$\frac{\partial y}{\partial u} = \frac{1}{v} = \frac{1}{r}.$$

However, if I want my two variables to be $u = r \theta$ and $v = \theta$, then $y = v$ and $$\frac{\partial y}{\partial u} = 0.$$

Partial derivative notation is ambiguous. In addition to specifying the variable that changes, it ought to also specify the variables that don't change, but for whatever reason it doesn't.

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