This section is completely optional.

 

Let y=f(x). Recall that we defined the derivative dydx(x0) as the limit of the increment quotient Δy/Δx when Δx approaches zero: dydx(x0)=limΔx0ΔyΔx, where Δy=y1y0 with y0=f(x0) and y1=f(x1)=f(x0+Δx).

We can think of the second derivative as the limit of the second increment quotient in the following manner. Consider x0,x1=x0+Δx,x2=x0+2Δx, and their corresponding y values, y0=f(x0),y1=f(x1), and y2=f(x2). We can take the second increment quotient as the first increment quotient of the first increment quotient; that is
(a)1Δx(y2y1Δxy1y0Δx)=1(Δx)2(y22y0+y1).
If we denote y1y0=Δy, and y2y1=Δy1, we can appropriately call y22y0+y1 , the increment of the increment of y, or the second increment of y and write symbolically y22y0+y1=Δy1Δy=Δ(Δy). We show the increment of the increment ΔΔ by Δ2, but we note that Δ2 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 Δ2y(Δx)2, where the numerator is the increment of the increment of y and the denominator is the square of Δx. The second derivative now is the limit of this quotient as Δx0. For higher derivatives, we can repeat this process. This symbolism made Leibniz introduce the following notations:
y=f(x)=d2ydx2,y=f(x)=d3ydx3,y(n)=f(n)(x)=dnydxn.

 

  • 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.