# Taking total differential of a nested function

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.

• 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 $$e=y$$)

• That's excellent. I still don't grasp it fully (as my math is super shaky still), though intuitively it makes a bit more sense. Thanks so much! – altec May 26 at 3:11