Let \(y=f(x)\). Recall that we defined the derivative \({\displaystyle \frac{dy}{dx}(x_{0})}\) as the limit of the increment quotient \(\Delta y/\Delta x\) when \(\Delta x\) approaches zero: \[\frac{dy}{dx}(x_{0})=\lim_{\Delta x\to0}\frac{\Delta y}{\Delta x},\] where \(\Delta y=y_{1}-y_{0}\) with \(y_{0}=f(x_{0})\) and \(y_{1}=f(x_{1})=f(x_{0}+\Delta x)\).

We can think of the second derivative as the limit of the second increment quotient in the following manner. Consider \(x_{0},x_{1}=x_{0}+\Delta x,x_{2}=x_{0}+2\Delta x\), and their corresponding \(y\) values, \(y_{0}=f(x_{0}),y_{1}=f(x_{1}),\) and \(y_{2}=f(x_{2})\). We can take the second increment quotient as the first increment quotient of the first increment quotient; that is

\[\frac{1}{\Delta x}\left(\frac{y_{2}-y_{1}}{\Delta x}-\frac{y_{1}-y_{0}}{\Delta x}\right)=\frac{1}{(\Delta x)^{2}}(y_{2}-2y_{0}+y_{1}).\tag{a}\]

If we denote \(y_{1}-y_{0}=\Delta y\), and \(y_{2}-y_{1}=\Delta y_{1}\), we can appropriately call \(y_{2}-2y_{0}+y_{1}\) , the increment of the increment of \(y\), or the second increment of \(y\) and write symbolically \[y_{2}-2y_{0}+y_{1}=\Delta y_{1}-\Delta y=\Delta(\Delta y).\] We show the increment of the increment \(\Delta\Delta\) by \(\Delta^{2}\), but we note that \(\Delta^{2}\) 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 \(\dfrac{\Delta^{2}y}{(\Delta x)^{2}}\), where the numerator is the increment of the increment of \(y\) and the denominator is the square of \(\Delta x\). The second derivative now is the limit of this quotient as \(\Delta x\to0\). For higher derivatives, we can repeat this process. This symbolism made Leibniz introduce the following notations:

\[y^{\prime\prime}=f^{\prime\prime}(x)=\frac{d^{2}y}{dx^{2}},\qquad y^{\prime\prime\prime}=f^{\prime\prime\prime}(x)=\frac{d^{3}y}{dx^{3}},\qquad y^{(n)}=f^{(n)}(x)=\frac{d^{n}y}{dx^{n}}.\]

- We already defined the second derivative as the limit of the first increment quotient of the first derivative (see Eq. (a) in Section 5.9). 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.