For example, I have the question "solve for $\frac {dy}{dx}$: $$ycos(xy)=y^2+2"$$ and I'm not sure if I should differentiate with the product rule or the chain rule first for the left side of the equation.
I apologize if this is a question with an obvious answer.
You would be taking a derivative of the type $$ \frac{d}{dx} \big[ y(x) \cdot g(x) \big] $$ for the function $$ g(x) = \cos \big( x \cdot y(x) \big) $$ Consequently, the natural first step is to apply the product rule. In doing so, you will need to find $g'(x)$, which will involve the chain rule.