Mnemonic For the Definition

The definition of the second differential can be remembered in the following way. By definition,

dy=f'(x)dx.

Now hold dx constant and formally apply the Constant Rule for differentiation, obtaining

d(dy)=f"(x)dx dx,

or

d²y = f "(x) dx².

(This is not a correct use of the Constant Rule because the rule applies to a real constant c, and dx is not a real number. It is only a mnemonic device to remember the definition of d²y, not a proof.)

The third and higher differentials can be motivated in the same way. If we hold dx constant and formally use the Constant Rule again and again, we obtain

and so on.