partiel derivative

That's a somewhat difficult and long explanation so as to really understand it. In a function of 1 variable y = f(x), if you go back to basic, basic fundamentals it all comes down to what the "diffrential of y" and the "differential of x" really mean and how both of them compare to f'(x). And why is it that when you perform the operation "dy by dx" you get the same answer when you find " the differential of y divided by the differential of x. The truth is that if you look at the definition of the "diffrential of x" there is NO requirement that it be infinitely small. So this leads to a conversation that takes up way too much space here to do it justice.

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Try this. Let's use a function of 2 variables. z = f(x,y). The total change in z (dz) is achieved by combining how much the change in x contributes to dz with how much the change in y contributes to dz.

pz/px ( y is held constant) gives you how much z has changed per unit of x( it's a rate). So multiplying by dx gives you the total change of z as contributed by x.

pz/px times dx

Similarly pz/py (x is held constant) gives you how much z has changed per unit of y. Multiplying by dy then gives you the the total change in z as contributed by y.

pz/py times dy

So when we combine them:

dz = (pz/px)dx + (py/pz)dy