In my 3rd year Microeconomics course we're deriving the Slutsky equation, and we have this general form at the start of the derivation:
$$x _ { l } ( p , e ( p , u ) ) = h _ { l } ( p , u )$$
And:
$$e ( p , u ) = y$$
Taking the total differential:
$$\biggr( \frac { \partial x _ { l } } { \partial p _ { l } } + \frac { \partial x _ { l } } { \partial y } \cdot \frac { \partial e } { \partial p _ { l } } \biggr) \mathrm dp_l = \frac { \partial h_l}{ \partial p_l}\cdot \mathrm dp_l $$
I'm unfamiliar with total differentials, though my understanding is it involves taking the derivative WRT all variables.
Could someone please advise:
why $\partial$ and $\mathrm d$ are being used simultaneously?
why the addition symbol is being used to split up terms on the LHS
confirmation of whether the chain rule is involved
For the same reason that we can write for $f(x,y,z)$ - $$df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy + \frac{\partial f}{\partial z}dz$$ Also, the chain rule has been used as - $$dx_l = \frac{\partial x_l}{\partial p_l}dp_l + \frac{\partial x_l}{\partial y}dy $$ $$\implies dx_l = \frac{\partial x_l}{\partial p_l}dp_l + \frac{\partial x_l}{\partial y}\frac{\partial e}{\partial p_l}dp_l$$ (as $e=y$)