partial derivative over product of multiple variables

Question

I was wondering how one would calculate a partial differential over a product for example $\frac{\partial xy^{2}}{\partial a}$ with $a=xy$ My issue being that defining $xy^{2}$ as $ay$ or $\frac{a^{2}}{x}$ will give different results.

Answer 1

Indeed, there are two answers. Here is a notation to reflect this: $$ \frac{\partial f}{\partial a}\bigg{|}_x \qquad \text{vs.} \qquad \frac{\partial f}{\partial a}\bigg{|}_y $$ Take $f=x$ as an example, $$ \frac{\partial x}{\partial a}\bigg{|}_x = 0 \qquad \text{vs.} \qquad \frac{\partial x}{\partial a}\bigg{|}_y = \frac{\partial}{\partial a}\bigg{|}_y\left( \frac{a}{y} \right) = \frac{1}{y}. $$