# Confusion regarding the Slutsky equation

I'm reading Henderson and Quandt's Microeconomic Theory textbook and in the derivation process of the slutsky equation the final formula confused me a bit. The first term on the right of the equation is said to represent the substition effect. It's mathematically represented as the partial derivative of q1 wrt p1 while utility is constant. Now this sounds contradictory to me, because partial of q1 wrt p1 implies q2, q3, q4..... remains constant, in which case the utility level shifts since only one quantity has changed. In order to hold utility constant, when one q1 changes other quantities have to change in order to hold utility constant, in which case the partial derivative makes no sense, because the idea of partial derivative implies everything else, including other quantities should be constant. If we hold utility constant the partial derivative doesnt make sense, and vice versa.

Am I missing something here? In my opinion the partial diff notation should be rejected in favor of the full differentiation with appropriate contraints as to what should be and shouldn't be.

The first term on the RHS that you are talking about is also the slope of the compensated demand curve aka Hicksian Demand curve which is a two-dimensional representation of the compensated demand function $$x^c=x^c(p_x,p_y,...,U)$$ which holds other prices and utility constant, or constrains the individual to stay on a particular indifference curve.