# Gottfried Wilhelm Leibniz

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Gottfried Wilhelm Leibniz (1646–1716) was a German polymath with multiple important contributions to mathematics and philosophy.

## Career and Contributions

Leibniz invented discovered calculus independently of Sir Isaac Newton. It is his notations (such as $\frac{dy}{dx}$ for differentials and the integral sign $\int\limits_a^b$) that are still used today. Newton's dot notation is more commonly used in physics to represent the time derivative. In fact, some expressions, such as the Euler-Lagrange equation, are most conveniently expressed using both Newton and Leibniz's notations. Also note that while the stylized s is universally used to represent an integral today, people still use different notations for the derivative. Joseph-Louis Lagrange used the prime notation while Leonard Euler used the letter D, treating it as an operator. However, more people associate calculus with Newton than with Leibniz[notes 1][notes 2]. Unfortunately, however, the two engaged in a prolonged and bitter but ultimately pointless priority dispute. We now know that Newton formulated his own version of calculus first, but published later as he despised controversies. Leibniz arrived at his later but published first. We also know from historical records that before the dispute, the two communicated with each other about Newton's work on the binomial theorem[1]. It is intriguing, but ultimately fruitless, to speculate on what might have been if the two cooperated and there was no priority disputes.

While Newton preferred to think of functions as infinite series to be differentiated or integrated term-by-term, Leibniz treated functions in a much less messy manner. As such, he discovered the Product and Quotient rules of differentiation. Starting from the Product Rule, one can use mathematical induction to arrive at the General Leibniz rule, which gives a higher-order derivative of a product of two function and is most conveniently expressed using the binomial notation. Knowing how to differentiate the products and quotients of functions, Leibniz moved on to composite functions and discovered the all-important Chain rule. Given the fact that he independently came up with the Fundamental Theorem of Calculus, it is no surprise that he also introduced the Leibniz Rule for Differentiating Under an Integral Sign, or simply the Leibniz Integral Rule.

Independent of James Bernoulli, Leibniz showed us how to solve what we now call separable ordinary differential equations (ODEs). His use of differentials definitely paved the way. Taking advantage of the Produce Rule, he introduced a technique to solve a slightly more complicated class of ODEs called first-order linear equations known as an integration factor. When James Bernoulli challenged other mathematicians to solve the brachistochrone problem, a classic in the calculus of variations, Leibniz responded with his own correct answer, independently of Newton and a few other mathematicians.

Leibniz also invented the binary system (01010100011001001110110110110000001),[notes 3] was a rationalist, and made contributions to medicine, philosophy, law, geology, and many other subjects.

## Notes

1. Because we have to use it to learn Newton's famous "physics" stuff.
2. Leibniz' followers in the very dismal, unproductive priority wars still had the last laugh. Leibniz' notation was immensely superior to Newton's and Leibniz attracted followers, including the Bernoulli family. The result was that British mathematics largely stagnated some quite some time whereas the continental mathematicians went from victory to victory.
3. Not to be confused with Morse Code.

## References

1. Stewart J. Calculus. Belmont, Cal.: Brooks/Cole, Cengage Learning; 2012.