How did Gottfried Wilhelm Leibniz use infinitesimals in calculus?

Gottfried Wilhelm Leibniz developed calculus using infinitesimals—quantities he described as infinitely small, yet not equal to zero—to perform operations on variables and functions.12

Leibniz treated continuous quantities as if they were discrete, applying his operators to these infinitesimals to create a powerful, general system of calculation:2

• Derivatives: He defined the derivative as a quotient of infinitesimals. By applying the difference operator d to a variable x, he obtained dx, an infinitesimal increase. The corresponding change in a function, df = f′dx, allowed him to express the derivative as the ratio f′ = df/dx.2
• Integrals: He viewed the integral, denoted by the symbol ∫, as a sum of infinitesimal strips of area under a curve. Consequently, he interpreted the fundamental theorem of calculus as the relationship where the difference between successive sums is the final term: d∫f(x)dx = f(x)dx.2

While his methods lacked rigorous logical justification and were famously criticized by George Berkeley as "ghosts of departed quantities," Leibniz’s apt notation and the undeniable utility of his approach were instrumental in the development of calculus.12