Leibniz's Notation

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.

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.

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:

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

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.