Differentials Discussion

Section 4.2 of our textbook introduces the related ideas of linear approximation and differentials. To my mind, introducing “differentials” as things that one can talk about in their own right is the most important part of this section, so this discussion is aimed at making the uses and limitations of that idea more concrete.

To start, we can follow the book’s idea of motivating differentials through linear approximations. In particular, the book’s linear approximation equation says that for values of x near some known value a,

Later, the book says essentially the same thing with differentials as

Can you explain in your own words how these 2 equations are saying roughly the same thing? Can you put into words any questions you have about the equations themselves (since they aren’t necessarily super self-explanatory)?