More about the Leibniz Notation for Higher Derivatives

Symbol	Meaning
$\mathrm{\Delta }y=y_1-y_0$	First increment of $y$
${\mathrm{\Delta }}^{2}y=y_2-2y_1+y_0$	Second increment of $y$
${\displaystyle \frac{{\mathrm{\Delta }}^{2}y}{(\mathrm{\Delta }x{)}^2}}$	Second-increment quotient
${\displaystyle \frac{d^{2}y}{d{x}^2}}$	Limit of second-increment quotient

The First Derivative as a Limit of Increment Quotients

Let $y=f(x)$. Recall:

$$\frac{dy}{dx}(x_0)=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}$$

where $\mathrm{\Delta }y=y_1-y_0$ with $y_0=f(x_0)$ and $y_1=f(x_0+\mathrm{\Delta }x)$.

The Second Derivative as a Limit of Second-Increment Quotients

Consider three equally spaced points:

$$x_0,x_1=x_0+\mathrm{\Delta }x,x_2=x_0+2\mathrm{\Delta }x$$

with corresponding values $y_0=f(x_0)$, $y_1=f(x_1)$, $y_2=f(x_2)$.

The second increment quotient is the "increment of the increment quotient":

\frac{1}{\Delta x}\left(\frac{y_2-y_1}{\Delta x}-\frac{y_1-y_0}{\Delta x}\right)=\frac{y_2-2y_1+y_0}{(\Delta x)^2}\tag{a}

We call $y_2-2y_1+y_0$ the second increment of $y$ and write it ${\mathrm{\Delta }}^{2}y$ (the symbol ${\mathrm{\Delta }}^2$ is not a square; it denotes the difference of differences).

The second-increment quotient is therefore ${\displaystyle \frac{{\mathrm{\Delta }}^{2}y}{(\mathrm{\Delta }x{)}^2}}$. The second derivative is the limit:

f''(x)=\lim_{\Delta x\to0}\frac{\Delta^2 y}{(\Delta x)^2}

(If the second derivative is continuous, this can be proved equivalent to the definition f''=(f')'.)

Leibniz's Notation

This symbolism motivated Leibniz to introduce:

y''=f''(x)=\frac{d^2y}{dx^2},\qquad y'''=f'''(x)=\frac{d^3y}{dx^3},\qquad y^{(n)}=f^{(n)}(x)=\frac{d^ny}{dx^n}

The numerator $d^{2}y$ suggests "increment of the increment of $y$" and the denominator $d{x}^2=(dx{)}^2$ is the square of the spacing.

Connection to Numerical Methods

Formula (a) is the basis of the finite difference method, a numerical technique for approximating differential equations. In that method:

f''(x)\approx\frac{f(x+h)-2f(x)+f(x-h)}{h^2}

for small step size $h$. This formula replaces the second derivative with computable arithmetic on function values and is widely used in engineering and scientific computing.

Notes

• The fact that the second derivative equals the limit of the second-increment quotient (not just the derivative of the first derivative) requires proof when the second derivative is continuous.
• The two definitions (limit of first derivative of first derivative, vs. limit of second-increment quotient) are equivalent whenever the second derivative is continuous.

Frequently Asked Questions

Why does $d^{2}y/d{x}^2$ look like a fraction but cannot be simplified to $dy/dx$ squared?

The notation is symbolic. $d^{2}y$ represents a second-order differential and $d{x}^2$ represents $(dx{)}^2$, not $d(x^2)$. The notation was chosen to echo the second-increment quotient ${\mathrm{\Delta }}^{2}y/(\mathrm{\Delta }x{)}^2$, but the $d$'s cannot be cancelled algebraically the way fractions can.

What is the finite difference formula for the second derivative?

For a function $f$ sampled at equally spaced points with spacing $h$: f''(x)\approx\dfrac{f(x+h)-2f(x)+f(x-h)}{h^2}. This is a central-difference approximation of order $h^2$.