How did Gottfried Wilhelm Leibniz’s notation shape modern calculus?

Gottfried Wilhelm Leibniz’s notation was instrumental in shaping modern calculus by providing a powerful, algorithmic symbolism that facilitated complex calculations. While Isaac Newton independently developed calculus, his own notation—based on "fluxions"—was less flexible and ultimately less influential in the long term.12

Leibniz introduced the symbols $d$ (for difference) and $\int$ (for sum), which allowed mathematicians to treat differentiation and integration as general, algebraic operations. Key aspects of his influence include:12

• Algorithmic Clarity: His notation, such as $df/dx$ for a derivative, made the process of differentiation explicit and easy to manipulate.13
• Symbolic Cancellation: His notation for the chain rule—expressed as $\frac{df}{dx}=\frac{df}{dg}·\frac{dg}{dx}$—functions with a memorable "symbolic cancellation" that remains a standard teaching tool today.3
• Generalization: Unlike Newton’s geometric approach, Leibniz’s notation allowed for the investigation of a wide class of functions, including transcendental curves, which helped establish calculus as a method of unparalleled scope.12

By creating a "perfectly apt notation," Leibniz enabled mathematicians to solve previously intractable problems, ensuring his symbolic system became the standard language of modern analysis.12