Gottfried Wilhelm Leibniz
| Thinking hard|
or hardly thinking?
|Major trains of thought|
|The good, the bad|
and the brain fart
|Come to think of it|
Career and contributions
invented[notes 1] discovered calculus independently of Sir Isaac Newton. As a consequence of his quest for the appropriate symbols to represent ideas, Leibniz introduced the notations (such as for differentials and for integrals) that are still used today. Newton's dot notation , on the other hand, is more commonly used in physics, for example, 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 , treating it as a linear operator. For partial derivatives, Gustav Jacob Jacobi employed the stylized d that resemble the number six in reverse , which may be thought of as an extension or modification of the Leibniz notation. Unfortunately, however, Newton and Leibniz 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, on the other hand, 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.  It is intriguing, but ultimately fruitless, to speculate on what might have been if the two cooperated and there were no priority disputes.
While Newton preferred to think of transcendental functions as infinite series to be differentiated or integrated term-by-term, Leibniz treated functions in a much less messy manner. Like Newton, Leibniz began by considering infinite series. But unlike Newton, whose starting materials were Euclidean, geometry, Cartesian geometry, and the work of John Wallis, Leibniz had his entryway via the differential triangle of Pascal. He unearthed the Product and Quotient rules of differentiation. One can generalize the Product Rule using 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. Newton independently discovered it via substitution. 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.
Alongside 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 Product 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. His work with infinite series led Leibniz to a beautiful[citation NOT needed] formula for  that we today typically derive using the Maclaurin series of the inverse tangent function.
- The truth is discovered, not invented.
- Not to be confused with Morse Code.
- Katz, Victor J. A History of Mathematics. 3rd ed., Addison-Wesley, 2009.
- Ball, W. W. Rouse. A Short Account of the History of Mathematics. 4th ed., Dover Publications, Inc., 1908.
- Stewart J. Calculus. Belmont, Cal.: Brooks/Cole, Cengage Learning; 2012.
- Bartle, Robert G., and Donald R. Sherbert. Introduction to Real Analysis. 4th ed., John Wiley & Sons, Inc., 2011.
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