Leibniz's Notation

GottfriedWho Wilhelmis Leibniz

Gottfried Wilhelm Leibniz (1646 - 1716) was a 17th century German mathematician. He’s often credited with developing many of the main principles of differential and integral calculus, and is primarily recognized for what we now call Leibniz’s notation.

LeibnizLeibniz's Notation System

The derivative of a function based on today’s standard is given by:

$\displaystyle\lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}=\displaystyle\lim_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$

Leibniz's notation expresses the derivative as:

$\frac{dy}{dx}=\frac{f(x+dx)-f(x)}{dx}$

where goes $dx$ toward 0.

Fractional Behavior

Let’s review some examples where Leibniz’s notation is often utilized. The Chain Rule using Lagrange Notation states:

$h^{'}(x)=f^{'}(g(x))g^{'}(x)$

We can translate the above Chain Rule into Leibniz's Notation as:

$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$

In the above equations, we can see how Leibniz’s Notation behaves similarly to a fraction, although it must be emphasized that the derivative is not a fraction.

Try It on a Function

Let us try $f(x)=x^{2}$ .

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Expand the function:

$\frac{dy}{dx}=\frac{(x+dx)^{2}-x^{2}}{dx}=\frac{x^{2}+2x(dx)+(dx)^{2}-x^{2}}{dx}$

Simplify fraction:

$\frac{dy}{dx}=\frac{2x(dx)+(dx)^{2}}{dx}=2x+dx=2x$

Try It on a Function

So the derivative of $x^{2}$ is $2x$ .

Leibniz first conceptualized $\frac{dy}{dx}$ as the quotient of an infinitely small change in y by an infinitely small change in x, called infinitesimals. However, this understanding lost popularity in the 19th-century when infinitesimals were considered too imprecise to define the infrastructure of calculus.

Leibniz’s original understanding of $\frac{dy}{dx}$ as a quotient has been reinterpreted to align with the modern limit-based definition of a derivative. Now, $dy$ and $dx$ are generally referred to as differentials instead of infinitesimals.