This section is completely optional.
Let . Recall that we defined the derivative as the limit of the increment quotient when approaches zero: where with and .
We can think of the second derivative as the limit of the second increment quotient in the following manner. Consider , and their corresponding values, and . We can take the second increment quotient as the first increment quotient of the first increment quotient; that is
If we denote , and , we can appropriately call , the increment of the increment of , or the second increment of and write symbolically We show the increment of the increment by , but we note that is not a square; it just denotes the increment of the increment (or the difference of the difference) or the second increment.
Using this symbolic notation, the second increment quotient is then , where the numerator is the increment of the increment of and the denominator is the square of . The second derivative now is the limit of this quotient as . For higher derivatives, we can repeat this process. This symbolism made Leibniz introduce the following notations:
- We already defined the second derivative as the limit of the first increment quotient of the first derivative (see Eq. (a) in the Section on Higher Derivatives). So the fact that the second derivative can also be represented as the limit of the second increment quotient needs proof. If the second derivative is continuous, it can be shown that these two definitions are equivalent.
- In finite difference method, a numerical method for solving equations, formula (a) is widely used to approximate the second derivative.