Geometrically, the derivative is the slope of the tangent line to the graph of f at a. The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function f. [49] :61–63 The earliest operation for curing stones is given in the Sushruta Samhita (6th century BCE). [2] The operation involved exposure and going up through the floor of the bladder. [2] The word comes from Latin calculus "small stone", from calx "limestone, lime", [3] probably related to Greek χάλιξ chalix "small stone, pebble, rubble", [4] which many trace to a Proto-Indo-European root for "split, break up". [5] Calculus was a term used for various kinds of stones. In the 18th century it came to be used for accidental or incidental mineral buildups in human and animal bodies, like kidney stones and minerals on teeth. [5] See also [ edit ] Some stone types (mainly those with substantial calcium content) can be detected on X-ray and CT scan When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first (later to be published in his Method of Fluxions), but Leibniz published his " Nova Methodus pro Maximis et Minimis" first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. This controversy divided English-speaking mathematicians from continental European mathematicians for many years, to the detriment of English mathematics. [32] A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus " the science of fluxions", a term that endured in English schools into the 19th century. [33] :100 The first complete treatise on calculus to be written in English and use the Leibniz notation was not published until 1815. [34] Maria Gaetana Agnesi

In kidney stones, calcium oxalate is the most common mineral type (see Nephrolithiasis). Uric acid is the second most common mineral type, but an in vitro study showed uric acid stones and crystals can promote the formation of calcium oxalate stones. [1] Pathophysiology [ edit ] During the Hellenistic period, this method was further developed by Archimedes ( c. 287 – c. 212 BC), who combined it with a concept of the indivisibles—a precursor to infinitesimals—allowing him to solve several problems now treated by integral calculus. In The Method of Mechanical Theorems he describes, for example, calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines. [9] ChinaMain article: Differential calculus Tangent line at ( x 0, f( x 0)). The derivative f′( x) of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point. Calculi of the gallbladder and bile ducts are called gallstones and are primarily developed from bile salts and cholesterol derivatives. Applications of differential calculus include computations involving velocity and acceleration, the slope of a curve, and optimization. [47] :341–453 Applications of integral calculus include computations involving area, volume, arc length, center of mass, work, and pressure. [47] :685–700 More advanced applications include power series and Fourier series.

Some stones can be directly recovered (at surgery, or when they leave the body spontaneously) and sent to a laboratory for analysis of content Here is a particular example, the derivative of the squaring function at the input 3. Let f( x) = x 2 be the squaring function. See also: Greek mathematics Archimedes used the method of exhaustion to calculate the area under a parabola in his work Quadrature of the Parabola. In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen ( c. 965– c. 1040 AD) derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid. [14] India If the input of the function represents time, then the derivative represents change concerning time. For example, if f is a function that takes time as input and gives the position of a ball at that time as output, then the derivative of f is how the position is changing in time, that is, it is the velocity of the ball. [31] :18–20This gives an exact value for the slope of a straight line. [49] :6 If the graph of the function is not a straight line, however, then the change in y divided by the change in x varies. Derivatives give an exact meaning to the notion of change in output concerning change in input. To be concrete, let f be a function, and fix a point a in the domain of f. ( a, f( a)) is a point on the graph of the function. If h is a number close to zero, then a + h is a number close to a. Therefore, ( a + h, f( a + h)) is close to ( a, f( a)). The slope between these two points is These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton. [29] He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for working with infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. Unlike Newton, Leibniz put painstaking effort into his choices of notation. [30] This expression is called a difference quotient. A line through two points on a curve is called a secant line, so m is the slope of the secant line between ( a, f( a)) and ( a + h, f( a + h)). The second line is only an approximation to the behavior of the function at the point a because it does not account for what happens between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero, which is undefined. The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero:

The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking. [46]There was a high probability of intraoperative and postoperative surgical complication like infection or bleeding Can progress to choledocholithiasis (gallstones in the bile duct) and gallstone pancreatitis (inflammation of the pancreas)

Calculi in the urinary system are called urinary calculi and include kidney stones (also called renal calculi or nephroliths) and bladder stones (also called vesical calculi or cystoliths). They can have any of several compositions, including mixed. Principal compositions include oxalate and urate.The derivative f′( x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of the second lines. Here the function involved (drawn in red) is f( x) = x 3 − x. The tangent line (in green) which passes through the point (−3/2, −15/8) has a slope of 23/4. The vertical and horizontal scales in this image are different. f ′ ( 3 ) = lim h → 0 ( 3 + h ) 2 − 3 2 h = lim h → 0 9 + 6 h + h 2 − 9 h = lim h → 0 6 h + h 2 h = lim h → 0 ( 6 + h ) = 6 {\displaystyle {\begin{aligned}f'(3)&=\lim _{h\to 0}{(3+h) A calculus ( pl.: calculi), often called a stone, is a concretion of material, usually mineral salts, that forms in an organ or duct of the body. Formation of calculi is known as lithiasis ( / ˌ l ɪ ˈ θ aɪ ə s ɪ s/). Stones can cause a number of medical conditions. Several mathematicians, including Maclaurin, tried to prove the soundness of using infinitesimals, but it would not be until 150 years later when, due to the work of Cauchy and Weierstrass, a way was finally found to avoid mere "notions" of infinitely small quantities. [38] The foundations of differential and integral calculus had been laid. In Cauchy's Cours d'Analyse, we find a broad range of foundational approaches, including a definition of continuity in terms of infinitesimals, and a (somewhat imprecise) prototype of an (ε, δ)-definition of limit in the definition of differentiation. [39] In his work Weierstrass formalized the concept of limit and eliminated infinitesimals (although his definition can validate nilsquare infinitesimals). Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called "infinitesimal calculus". Bernhard Riemann used these ideas to give a precise definition of the integral. [40] It was also during this period that the ideas of calculus were generalized to the complex plane with the development of complex analysis. [41] Obstruction of an opening or duct, interfering with normal flow and disrupting the function of the organ in question