How to derive substitution and income effect using Slutsky equation if we don't know which of the prices change?

Question

We have the utility function $$U(x,y)=x + y$$ and we have to derive the substitution and income effects using Slutsky equation. But after I derive the Hicksian demand functions for e.g. x:

$$h_x= \frac{I}{p_x+\frac{p_y^3}{p_x^2}}$$

do we derive this only with respect to x in order to account for impacts of changes in $p_x$ or do I do the same derivation with respect to y and sum both up?

The confusing part is that I don't know what prices change, so it would seem most generally to assume that both change.

Answer 1

Probably just an issue with the phrasing of the problem you're tackling. I'd assume they just want you be using the Slutsky equation for own-price changes, so you should be good just differentiating what you derived with respect to px and then carrying on with the income effect.

Answer 2

Just take the total derivative:

$Dh_x = \frac{dh_x}{dp_x}dp_x + \frac{dh_x}{dp_y}dp_y$

Notice that this "contains" both partial derivatives, which, by definition, are each defined only when the other variable is held constant (e.g., $p_y = 0$).