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111-414: In calculus , the second derivative , or the second-order derivative , of a function f is the derivative of the derivative of f . Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of

222-500: A ) + f ′ ( a ) ( x − a ) + 1 2 f ″ ( a ) ( x − a ) 2 . {\displaystyle f(x)\approx f(a)+f'(a)(x-a)+{\tfrac {1}{2}}f''(a)(x-a)^{2}.} This quadratic approximation is the second-order Taylor polynomial for the function centered at x = a . For many combinations of boundary conditions explicit formulas for eigenvalues and eigenvectors of

333-448: A vinculum to indicate grouping of symbols, but later he introduced the idea of using pairs of parentheses for this purpose, thus appeasing the typesetters who no longer had to widen the spaces between lines on a page and making the pages look more attractive. Many of the over 200 new symbols introduced by Leibniz are still in use today. Besides the differentials dx , dy and the integral sign ( ∫ ) already mentioned, he also introduced

444-482: A Leibniz-like development of the usual rules of calculus. There is also smooth infinitesimal analysis , which differs from non-standard analysis in that it mandates neglecting higher-power infinitesimals during derivations. Based on the ideas of F. W. Lawvere and employing the methods of category theory , smooth infinitesimal analysis views all functions as being continuous and incapable of being expressed in terms of discrete entities. One aspect of this formulation

555-509: 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. 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

666-469: A continuous version of the second difference for sequences . However, the existence of the above limit does not mean that the function f {\displaystyle f} has a second derivative. The limit above just gives a possibility for calculating the second derivative—but does not provide a definition. A counterexample is the sign function sgn ⁡ ( x ) {\displaystyle \operatorname {sgn}(x)} , which

777-412: A differential, as it might appear in an arc length formula for instance, was written as dxdx . However, Leibniz did use his d notation as we would today use operators, namely he would write a second derivative as ddy and a third derivative as dddy . In 1695 Leibniz started to write d ⋅ x and d ⋅ x for ddx and dddx respectively, but l'Hôpital , in his textbook on calculus written around

888-479: A finite difference). The expression may also be thought of as the application of the differential operator ⁠ d / dx ⁠ (again, a single symbol) to y , regarded as a function of x . This operator is written D in Euler's notation . Leibniz did not use this form, but his use of the symbol d corresponds fairly closely to this modern concept. While there is traditionally no division implied by

999-405: A fluctuating velocity over a given period. If f ( x ) represents speed as it varies over time, the distance traveled between the times represented by a and b is the area of the region between f ( x ) and the x -axis, between x = a and x = b . Leibniz%27s notation Consider y as a function of a variable x , or y = f ( x ) . If this is the case, then

1110-454: A foundation of calculus, was eventually replaced by rigorous concepts developed by Weierstrass and others in the 19th century. Consequently, Leibniz's quotient notation was re-interpreted to stand for the limit of the modern definition. However, in many instances, the symbol did seem to act as an actual quotient would and its usefulness kept it popular even in the face of several competing notations. Several different formalisms were developed in

1221-463: A function f ( x ) {\displaystyle f(x)} is usually denoted f ″ ( x ) {\displaystyle f''(x)} . That is: f ″ = ( f ′ ) ′ {\displaystyle f''=\left(f'\right)'} When using Leibniz's notation for derivatives, the second derivative of a dependent variable y with respect to an independent variable x

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1332-457: A function f can be used to determine the concavity of the graph of f . A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function. Similarly, a function whose second derivative is negative will be concave down (sometimes simply called concave), and its tangent line will lie above

1443-580: A function is equal to the divergence of the gradient , and the trace of the Hessian matrix. Calculus Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations . Originally called infinitesimal calculus or "the calculus of infinitesimals ", it has two major branches, differential calculus and integral calculus . The former concerns instantaneous rates of change , and

1554-512: A method that would later be called Cavalieri's principle to find the volume of a sphere . 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

1665-691: A minority of mathematicians. Jerome Keisler wrote a first-year calculus textbook, Elementary calculus: an infinitesimal approach , based on Robinson's approach. From the point of view of modern infinitesimal theory, Δ x is an infinitesimal x -increment, Δ y is the corresponding y -increment, and the derivative is the standard part of the infinitesimal ratio: Then one sets d x = Δ x {\displaystyle dx=\Delta x} , d y = f ′ ( x ) d x {\displaystyle dy=f'(x)dx} , so that by definition, f ′ ( x ) {\displaystyle f'(x)}

1776-435: A more rigorous foundation for calculus, and for this reason, they became the standard approach during the 20th century. However, the infinitesimal concept was revived in the 20th century with the introduction of non-standard analysis and smooth infinitesimal analysis , which provided solid foundations for the manipulation of infinitesimals. Differential calculus is the study of the definition, properties, and applications of

1887-412: A quotient of differentials , this should not be done with the higher order forms. However, an alternative Leibniz notation for differentiation for higher orders allows for this. This notation was, however, not used by Leibniz. In print he did not use multi-tiered notation nor numerical exponents (before 1695). To write x for instance, he would write xxx , as was common in his time. The square of

1998-415: A real-world analogy. Consider a vehicle that at first is moving forward at a great velocity, but with a negative acceleration. Clearly, the position of the vehicle at the point where the velocity reaches zero will be the maximum distance from the starting position – after this time, the velocity will become negative and the vehicle will reverse. The same is true for the minimum, with a vehicle that at first has

2109-473: A specific value. However, the Leibniz notation has other virtues that have kept it popular through the years. In its modern interpretation, the expression ⁠ dy / dx ⁠ should not be read as the division of two quantities dx and dy (as Leibniz had envisioned it); rather, the whole expression should be seen as a single symbol that is shorthand for (note Δ vs. d , where Δ indicates

2220-470: A steady 50 mph for 3 hours results in a total distance of 150 miles. Plotting the velocity as a function of time yields a rectangle with a height equal to the velocity and a width equal to the time elapsed. Therefore, the product of velocity and time also calculates the rectangular area under the (constant) velocity curve. This connection between the area under a curve and the distance traveled can be extended to any irregularly shaped region exhibiting

2331-407: A straight line), then the function can be written as y = mx + b , where x is the independent variable, y is the dependent variable, b is the y -intercept, and: This gives an exact value for the slope of a straight line. 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

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2442-455: A very negative velocity but positive acceleration. It is possible to write a single limit for the second derivative: f ″ ( x ) = lim h → 0 f ( x + h ) − 2 f ( x ) + f ( x − h ) h 2 . {\displaystyle f''(x)=\lim _{h\to 0}{\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}.} The limit

2553-401: Is Lagrange's notation Another alternative is Newton's notation , often used for derivatives with respect to time (like velocity ), which requires placing a dot over the dependent variable (in this case, x ): Lagrange's " prime " notation is especially useful in discussions of derived functions and has the advantage of having a natural way of denoting the value of the derived function at

2664-417: Is also used to gain a more precise understanding of the nature of space, time, and motion. For centuries, mathematicians and philosophers wrestled with paradoxes involving division by zero or sums of infinitely many numbers. These questions arise in the study of motion and area. The ancient Greek philosopher Zeno of Elea gave several famous examples of such paradoxes . Calculus provides tools, especially

2775-416: Is an abbreviation of both infinitesimal calculus and integral calculus , which denotes courses of elementary mathematical analysis . In Latin , the word calculus means “small pebble”, (the diminutive of calx , meaning "stone"), a meaning which still persists in medicine . Because such pebbles were used for counting out distances, tallying votes, and doing abacus arithmetic, the word came to be

2886-404: 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

2997-759: Is called the second symmetric derivative . The second symmetric derivative may exist even when the (usual) second derivative does not. The expression on the right can be written as a difference quotient of difference quotients: f ( x + h ) − 2 f ( x ) + f ( x − h ) h 2 = f ( x + h ) − f ( x ) h − f ( x ) − f ( x − h ) h h . {\displaystyle {\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}={\frac {{\dfrac {f(x+h)-f(x)}{h}}-{\dfrac {f(x)-f(x-h)}{h}}}{h}}.} This limit can be viewed as

3108-416: Is common in calculus.) The definite integral inputs a function and outputs a number, which gives the algebraic sum of areas between the graph of the input and the x-axis . The technical definition of the definite integral involves the limit of a sum of areas of rectangles, called a Riemann sum . A motivating example is the distance traveled in a given time. If the speed is constant, only multiplication

3219-434: Is defined as: sgn ⁡ ( x ) = { − 1 if  x < 0 , 0 if  x = 0 , 1 if  x > 0. {\displaystyle \operatorname {sgn}(x)={\begin{cases}-1&{\text{if }}x<0,\\0&{\text{if }}x=0,\\1&{\text{if }}x>0.\end{cases}}} The sign function

3330-404: Is developed using limits rather than infinitesimals, it is common to manipulate symbols like dx and dy as if they were real numbers; although it is possible to avoid such manipulations, they are sometimes notationally convenient in expressing operations such as the total derivative . Integral calculus is the study of the definitions, properties, and applications of two related concepts,

3441-417: Is differentiable at x and its derivative can be expressed in Leibniz notation as, This can be generalized to deal with the composites of several appropriately defined and related functions, u 1 , u 2 , ..., u n and would be expressed as, Also, the integration by substitution formula may be expressed by where x is thought of as a function of a new variable u and the function y on

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3552-651: 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. Applications of differential calculus include computations involving velocity and acceleration , the slope of a curve, and optimization . Applications of integral calculus include computations involving area, volume , arc length , center of mass , work , and pressure . More advanced applications include power series and Fourier series . Calculus

3663-418: Is in harmony with dimensional analysis . Suppose a dependent variable y represents a function f of an independent variable x , that is, Then the derivative of the function f , in Leibniz's notation for differentiation , can be written as The Leibniz expression, also, at times, written dy / dx , is one of several notations used for derivatives and derived functions. A common alternative

3774-416: Is needed: But if the speed changes, a more powerful method of finding the distance is necessary. One such method is to approximate the distance traveled by breaking up the time into many short intervals of time, then multiplying the time elapsed in each interval by one of the speeds in that interval, and then taking the sum (a Riemann sum ) of the approximate distance traveled in each interval. The basic idea

3885-1343: Is not continuous at zero, and therefore the second derivative for x = 0 {\displaystyle x=0} does not exist. But the above limit exists for x = 0 {\displaystyle x=0} : lim h → 0 sgn ⁡ ( 0 + h ) − 2 sgn ⁡ ( 0 ) + sgn ⁡ ( 0 − h ) h 2 = lim h → 0 sgn ⁡ ( h ) − 2 ⋅ 0 + sgn ⁡ ( − h ) h 2 = lim h → 0 sgn ⁡ ( h ) + ( − sgn ⁡ ( h ) ) h 2 = lim h → 0 0 h 2 = 0. {\displaystyle {\begin{aligned}\lim _{h\to 0}{\frac {\operatorname {sgn}(0+h)-2\operatorname {sgn}(0)+\operatorname {sgn}(0-h)}{h^{2}}}&=\lim _{h\to 0}{\frac {\operatorname {sgn}(h)-2\cdot 0+\operatorname {sgn}(-h)}{h^{2}}}\\&=\lim _{h\to 0}{\frac {\operatorname {sgn}(h)+(-\operatorname {sgn}(h))}{h^{2}}}=\lim _{h\to 0}{\frac {0}{h^{2}}}=0.\end{aligned}}} Just as

3996-447: Is still to some extent an active area of research today. 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. The foundations of differential and integral calculus had been laid. In Cauchy's Cours d'Analyse , we find

4107-407: Is that if only a short time elapses, then the speed will stay more or less the same. However, a Riemann sum only gives an approximation of the distance traveled. We must take the limit of all such Riemann sums to find the exact distance traveled. When velocity is constant, the total distance traveled over the given time interval can be computed by multiplying velocity and time. For example, traveling

4218-421: Is that the law of excluded middle does not hold. The law of excluded middle is also rejected in constructive mathematics , a branch of mathematics that insists that proofs of the existence of a number, function, or other mathematical object should give a construction of the object. Reformulations of calculus in a constructive framework are generally part of the subject of constructive analysis . While many of

4329-418: Is the integral in which the infinitesimal increments are summed (e.g. to compute lengths, areas and volumes as sums of tiny pieces), for which Leibniz also supplied a closely related notation involving the same differentials, a notation whose efficiency proved decisive in the development of continental European mathematics. Leibniz's concept of infinitesimals, long considered to be too imprecise to be used as

4440-410: Is the doubling function. A common notation, introduced by Leibniz, for the derivative in the example above is In an approach based on limits, the symbol ⁠ dy / dx ⁠ is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above. Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, dy being

4551-462: Is the ratio of dy by dx . Similarly, although most mathematicians now view an integral as a limit where Δ x is an interval containing x i , Leibniz viewed it as the sum (the integral sign denoted summation for him) of infinitely many infinitesimal quantities f ( x )  dx . From the viewpoint of nonstandard analysis, it is correct to view the integral as the standard part of such an infinite sum. The trade-off needed to gain

Second derivative - Misplaced Pages Continue

4662-427: Is the second derivative of position ( x ) with respect to time. On the graph of a function , the second derivative corresponds to the curvature or concavity of the graph. The graph of a function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the opposite way. The power rule for the first derivative, if applied twice, will produce

4773-468: Is written d 2 y d x 2 . {\displaystyle {\frac {d^{2}y}{dx^{2}}}.} This notation is derived from the following formula: d 2 y d x 2 = d d x ( d y d x ) . {\displaystyle {\frac {d^{2}y}{dx^{2}}}\,=\,{\frac {d}{dx}}\left({\frac {dy}{dx}}\right).} Given

4884-421: Is zero is necessarily a point of inflection. The relation between the second derivative and the graph can be used to test whether a stationary point for a function (i.e., a point where f ′ ( x ) = 0 {\displaystyle f'(x)=0} ) is a local maximum or a local minimum . Specifically, The reason the second derivative produces these results can be seen by way of

4995-537: The dy s and dx s as separable. One of the simplest types of differential equations is where M and N are continuous functions. Solving (implicitly) such an equation can be done by examining the equation in its differential form , and integrating to obtain Rewriting, when possible, a differential equation into this form and applying the above argument is known as the separation of variables technique for solving such equations. In each of these instances

5106-456: 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 . The method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD to find the area of a circle. In the 5th century AD, Zu Gengzhi , son of Zu Chongzhi , established

5217-400: The derivative of y with respect to x , which later came to be viewed as the limit was, according to Leibniz, the quotient of an infinitesimal increment of y by an infinitesimal increment of x , or where the right hand side is Joseph-Louis Lagrange's notation for the derivative of f at x . The infinitesimal increments are called differentials . Related to this

5328-403: The derivative of a function. The process of finding the derivative is called differentiation . Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the derivative function or just

5439-417: The derivative of the original function. In formal terms, the derivative is a linear operator which takes a function as its input and produces a second function as its output. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if

5550-403: The indefinite integral and the definite integral . The process of finding the value of an integral is called integration . The indefinite integral, also known as the antiderivative , is the inverse operation to the derivative. F is an indefinite integral of f when f is a derivative of F . (This use of lower- and upper-case letters for a function and its indefinite integral

5661-404: The limit and the infinite series , that resolve the paradoxes. Calculus is usually developed by working with very small quantities. Historically, the first method of doing so was by infinitesimals . These are objects which can be treated like real numbers but which are, in some sense, "infinitely small". For example, an infinitesimal number could be greater than 0, but less than any number in

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5772-512: The method of exhaustion to prove the formulas for cone and pyramid volumes. 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

5883-449: The n th derivative of f in Leibniz notation is given by, This notation, for the second derivative , is obtained by using ⁠ d / dx ⁠ as an operator in the following way, A third derivative, which might be written as, can be obtained from Similarly, the higher derivatives may be obtained inductively. While it is possible, with carefully chosen definitions, to interpret ⁠ dy / dx ⁠ as

5994-488: The slopes of curves , while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus . They make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit . It is the "mathematical backbone" for dealing with problems where variables change with time or some other reference variable. Infinitesimal calculus

6105-545: The 20th century that can give rigorous meaning to notions of infinitesimals and infinitesimal displacements, including nonstandard analysis , tangent space , O notation and others. The derivatives and integrals of calculus can be packaged into the modern theory of differential forms , in which the derivative is genuinely a ratio of two differentials, and the integral likewise behaves in exact accordance with Leibniz notation. However, this requires that derivative and integral first be defined by other means, and as such expresses

6216-665: The Latin word for calculation . In this sense, it was used in English at least as early as 1672, several years before the publications of Leibniz and Newton, who wrote their mathematical texts in Latin. In addition to differential calculus and integral calculus, the term is also used for naming specific methods of computation or theories that imply some sort of computation. Examples of this usage include propositional calculus , Ricci calculus , calculus of variations , lambda calculus , sequent calculus , and process calculus . Furthermore,

6327-493: The Leibniz notation for a derivative appears to act like a fraction, even though, in its modern interpretation, it isn't one. In the 1960s, building upon earlier work by Edwin Hewitt and Jerzy Łoś , Abraham Robinson developed mathematical explanations for Leibniz's infinitesimals that were acceptable by contemporary standards of rigor, and developed nonstandard analysis based on these ideas. Robinson's methods are used by only

6438-439: The Leibniz notation was not published until 1815. Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on both infinitesimal and integral calculus was written in 1748 by Maria Gaetana Agnesi . In calculus, foundations refers to the rigorous development of the subject from axioms and definitions. In early calculus,

6549-558: The Middle East, and still later again in medieval Europe and India. Calculations of volume and area , one goal of integral calculus, can be found in the Egyptian Moscow papyrus ( c.  1820   BC ), but the formulae are simple instructions, with no indication as to how they were obtained. Laying the foundations for integral calculus and foreshadowing the concept of the limit, ancient Greek mathematician Eudoxus of Cnidus ( c.  390–337   BC ) developed

6660-600: The article " Nova Methodus pro Maximis et Minimis " also published in Acta Eruditorum in 1684. While the symbol ⁠ dx / dy ⁠ does appear in private manuscripts of 1675, it does not appear in this form in either of the above-mentioned published works. Leibniz did, however, use forms such as dy ad dx and dy  : dx in print. At the end of the 19th century, Weierstrass's followers ceased to take Leibniz's notation for derivatives and integrals literally. That is, mathematicians felt that

6771-459: The available signs, and you will observe, Sir, by the small enclosure [on determinants] that Vieta and Descartes have not known all the mysteries. He refined his criteria for good notation over time and came to realize the value of "adopting symbolisms which could be set up in a line like ordinary type, without the need of widening the spaces between lines to make room for symbols with sprawling parts." For instance, in his early works he heavily used

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6882-403: 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: Geometrically, the derivative is the slope of the tangent line to

6993-413: The concept of infinitesimals contained logical contradictions in its development. A number of 19th century mathematicians (Weierstrass and others) found logically rigorous ways to treat derivatives and integrals without infinitesimals using limits as shown above, while Cauchy exploited both infinitesimals and limits (see Cours d'Analyse ). Nonetheless, Leibniz's notation is still in general use. Although

7104-692: The corresponding eigenvectors (also called eigenfunctions ) are v j ( x ) = 2 L sin ⁡ ( j π x L ) {\displaystyle v_{j}(x)={\sqrt {\tfrac {2}{L}}}\sin \left({\tfrac {j\pi x}{L}}\right)} . Here, v j ″ ( x ) = λ j v j ( x ) {\displaystyle v''_{j}(x)=\lambda _{j}v_{j}(x)} , for j = 1 , … , ∞ {\displaystyle j=1,\ldots ,\infty } . For other well-known cases, see Eigenvalues and eigenvectors of

7215-485: The detriment of English mathematics. 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. The first complete treatise on calculus to be written in English and use

7326-590: The differential of y ranged successively from ω , l , and ⁠ y / d ⁠ until he finally settled on dy . His integral sign first appeared publicly in the article " De Geometria Recondita et analysi indivisibilium atque infinitorum " ("On a hidden geometry and analysis of indivisibles and infinites"), published in Acta Eruditorum in June 1686, but he had been using it in private manuscripts at least since 1675. Leibniz first used dx in

7437-464: The discovery that cosine is the derivative of sine . In the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions. Madhava of Sangamagrama and the Kerala School of Astronomy and Mathematics stated components of calculus, but according to Victor J. Katz they were not able to "combine many differing ideas under

7548-421: The first derivative is related to linear approximations , the second derivative is related to the best quadratic approximation for a function f . This is the quadratic function whose first and second derivatives are the same as those of f at a given point. The formula for the best quadratic approximation to a function f around the point x = a is f ( x ) ≈ f (

7659-399: The foundation of calculus. Another way is to use Abraham Robinson 's non-standard analysis . Robinson's approach, developed in the 1960s, uses technical machinery from mathematical logic to augment the real number system with infinitesimal and infinite numbers, as in the original Newton-Leibniz conception. The resulting numbers are called hyperreal numbers , and they can be used to give

7770-526: The function f ( x ) = x 3 , {\displaystyle f(x)=x^{3},} the derivative of f is the function f ′ ( x ) = 3 x 2 . {\displaystyle f'(x)=3x^{2}.} The second derivative of f is the derivative of f ′ {\displaystyle f'} , namely f ″ ( x ) = 6 x . {\displaystyle f''(x)=6x.} The second derivative of

7881-406: The function g ( x ) = 2 x , as will turn out. In Lagrange's notation , the symbol for a derivative is an apostrophe -like mark called a prime . Thus, the derivative of a function called f is denoted by f′ , pronounced "f prime" or "f dash". For instance, if f ( x ) = x is the squaring function, then f′ ( x ) = 2 x is its derivative (the doubling function g from above). If

7992-1145: The function's image and domain both have a potential, then these fit together into a symmetric matrix known as the Hessian . The eigenvalues of this matrix can be used to implement a multivariable analogue of the second derivative test. (See also the second partial derivative test .) Another common generalization of the second derivative is the Laplacian . This is the differential operator ∇ 2 {\displaystyle \nabla ^{2}} (or Δ {\displaystyle \Delta } ) defined by ∇ 2 f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 . {\displaystyle \nabla ^{2}f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}.} The Laplacian of

8103-415: 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 . Here is a particular example, the derivative of the squaring function at the input 3. Let f ( x ) = x be the squaring function. The slope of the tangent line to the squaring function at

8214-407: The graph of the function near the point of contact. If the second derivative of a function changes sign, the graph of the function will switch from concave down to concave up, or vice versa. A point where this occurs is called an inflection point . Assuming the second derivative is continuous, it must take a value of zero at any inflection point, although not every point where the second derivative

8325-513: The ideas of calculus had been developed earlier in Greece , China , India , Iraq, Persia , and Japan , the use of calculus began in Europe, during the 17th century, when Newton and Leibniz built on the work of earlier mathematicians to introduce its basic principles. The Hungarian polymath John von Neumann wrote of this work, The calculus was the first achievement of modern mathematics and it

8436-456: The infinitesimally small change in y caused by an infinitesimally small change dx applied to x . We can also think of ⁠ d / dx ⁠ as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example: In this usage, the dx in the denominator is read as "with respect to x ". Another example of correct notation could be: Even when calculus

8547-424: The initial elongated s of the Latin word ſ umma ("sum") as written at the time. Viewing differences as the inverse operation of summation, he used the symbol d , the first letter of the Latin differentia , to indicate this inverse operation. Leibniz was fastidious about notation, having spent years experimenting, adjusting, rejecting and corresponding with other mathematicians about them. Notations he used for

8658-401: 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. If a function is linear (that is if the graph of the function is

8769-417: The intrinsic structure of the real number system (as a metric space with the least-upper-bound property ). In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by sequences of smaller and smaller numbers, and the infinitely small behavior of a function is found by taking the limiting behavior for these sequences. Limits were thought to provide

8880-410: The left is expressed in terms of x while on the right it is expressed in terms of u . If y = f ( x ) where f is a differentiable function that is invertible , the derivative of the inverse function, if it exists, can be given by, where the parentheses are added to emphasize the fact that the derivative is not a fraction. However, when solving differential equations, it is easy to think of

8991-536: The mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid , and many other problems discussed in his Principia Mathematica (1687). In other work, he developed series expansions for functions, including fractional and irrational powers, and it

9102-520: The mixed partials ∂ 2 f ∂ x ∂ y , ∂ 2 f ∂ x ∂ z ,  and  ∂ 2 f ∂ y ∂ z . {\displaystyle {\frac {\partial ^{2}f}{\partial x\,\partial y}},\;{\frac {\partial ^{2}f}{\partial x\,\partial z}},{\text{ and }}{\frac {\partial ^{2}f}{\partial y\,\partial z}}.} If

9213-404: The notation (but see Nonstandard analysis ), the division-like notation is useful since in many situations, the derivative operator does behave like a division, making some results about derivatives easy to obtain and remember. This notation owes its longevity to the fact that it seems to reach to the very heart of the geometrical and mechanical applications of the calculus. If y = f ( x ) ,

9324-412: The notation need not be taken literally, it is usually simpler than alternatives when the technique of separation of variables is used in the solution of differential equations. In physical applications, one may for example regard f ( x ) as measured in meters per second, and d x in seconds, so that f ( x ) d x is in meters, and so is the value of its definite integral. In that way the Leibniz notation

9435-413: 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 This expression

9546-537: The object is changing with respect to time. In Leibniz notation : a = d v d t = d 2 x d t 2 , {\displaystyle a={\frac {dv}{dt}}={\frac {d^{2}x}{dt^{2}}},} where a is acceleration, v is velocity, t is time, x is position, and d is the instantaneous "delta" or change. The last expression d 2 x d t 2 {\displaystyle {\tfrac {d^{2}x}{dt^{2}}}}

9657-419: The point (3, 9) is 6, that is to say, it is going up six times as fast as it is going to the right. The limit process just described can be performed for any point in the domain of the squaring function. This defines the derivative function of the squaring function or just the derivative of the squaring function for short. A computation similar to the one above shows that the derivative of the squaring function

9768-430: The precision of these concepts is that the set of real numbers must be extended to the set of hyperreal numbers . Leibniz experimented with many different notations in various areas of mathematics. He felt that good notation was fundamental in the pursuit of mathematics. In a letter to l'Hôpital in 1693 he says: One of the secrets of analysis consists in the characteristic, that is, in the art of skilful employment of

9879-418: The same time, used Leibniz's original forms. One reason that Leibniz's notations in calculus have endured so long is that they permit the easy recall of the appropriate formulas used for differentiation and integration. For instance, the chain rule —suppose that the function g is differentiable at x and y = f ( u ) is differentiable at u = g ( x ) . Then the composite function y = f ( g ( x ))

9990-581: The second derivative can be obtained. For example, assuming x ∈ [ 0 , L ] {\displaystyle x\in [0,L]} and homogeneous Dirichlet boundary conditions (i.e., v ( 0 ) = v ( L ) = 0 {\displaystyle v(0)=v(L)=0} where v is the eigenvector), the eigenvalues are λ j = − j 2 π 2 L 2 {\displaystyle \lambda _{j}=-{\tfrac {j^{2}\pi ^{2}}{L^{2}}}} and

10101-662: The second derivative . The second derivative generalizes to higher dimensions through the notion of second partial derivatives . For a function f : R → R , these include the three second-order partials ∂ 2 f ∂ x 2 , ∂ 2 f ∂ y 2 ,  and  ∂ 2 f ∂ z 2 {\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}},\;{\frac {\partial ^{2}f}{\partial y^{2}}},{\text{ and }}{\frac {\partial ^{2}f}{\partial z^{2}}}} and

10212-646: The second derivative power rule as follows: d 2 d x 2 x n = d d x d d x x n = d d x ( n x n − 1 ) = n d d x x n − 1 = n ( n − 1 ) x n − 2 . {\displaystyle {\frac {d^{2}}{dx^{2}}}x^{n}={\frac {d}{dx}}{\frac {d}{dx}}x^{n}={\frac {d}{dx}}\left(nx^{n-1}\right)=n{\frac {d}{dx}}x^{n-1}=n(n-1)x^{n-2}.} The second derivative of

10323-442: The self-consistency and computational efficacy of the Leibniz notation rather than giving it a new foundation. The Newton–Leibniz approach to infinitesimal calculus was introduced in the 17th century. While Newton worked with fluxions and fluents, Leibniz based his approach on generalizations of sums and differences. Leibniz adapted the integral symbol ∫ {\displaystyle \textstyle \int } from

10434-410: The sequence 1, 1/2, 1/3, ... and thus less than any positive real number . From this point of view, calculus is a collection of techniques for manipulating infinitesimals. The symbols d x {\displaystyle dx} and d y {\displaystyle dy} were taken to be infinitesimal, and the derivative d y / d x {\displaystyle dy/dx}

10545-408: The squaring function is given the input three, then it outputs nine. The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function. The function produced by differentiating

10656-472: The squaring function turns out to be the doubling function. In more explicit terms the "doubling function" may be denoted by g ( x ) = 2 x and the "squaring function" by f ( x ) = x . The "derivative" now takes the function f ( x ) , defined by the expression " x ", as an input, that is all the information—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to output another function,

10767-439: The subject is still occasionally called "infinitesimal calculus". Bernhard Riemann used these ideas to give a precise definition of the integral. It was also during this period that the ideas of calculus were generalized to the complex plane with the development of complex analysis . In modern mathematics, the foundations of calculus are included in the field of real analysis , which contains full definitions and proofs of

10878-467: The term "calculus" has variously been applied in ethics and philosophy, for such systems as Bentham's felicific calculus , and the ethical calculus . Modern calculus was developed in 17th-century Europe by Isaac Newton and Gottfried Wilhelm Leibniz (independently of each other, first publishing around the same time) but elements of it first appeared in ancient Egypt and later Greece, then in China and

10989-414: The theorems of calculus. The reach of calculus has also been greatly extended. Henri Lebesgue invented measure theory , based on earlier developments by Émile Borel , and used it to define integrals of all but the most pathological functions. Laurent Schwartz introduced distributions , which can be used to take the derivative of any function whatsoever. Limits are not the only rigorous approach to

11100-425: The two unifying themes of the derivative and the integral , show the connection between the two, and turn calculus into the great problem-solving tool we have today". Johannes Kepler 's work Stereometria Doliorum (1615) formed the basis of integral calculus. Kepler developed a method to calculate the area of an ellipse by adding up the lengths of many radii drawn from a focus of the ellipse. Significant work

11211-408: The use of infinitesimal quantities was thought unrigorous and was fiercely criticized by several authors, most notably Michel Rolle and Bishop Berkeley . Berkeley famously described infinitesimals as the ghosts of departed quantities in his book The Analyst in 1734. Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz, and

11322-687: The volume of a paraboloid . Bhāskara II ( c.  1114–1185 ) was acquainted with some ideas of differential calculus and suggested that the "differential coefficient" vanishes at an extremum value of the function. In his astronomical work, he gave a procedure that looked like a precursor to infinitesimal methods. Namely, if x ≈ y {\displaystyle x\approx y} then sin ⁡ ( y ) − sin ⁡ ( x ) ≈ ( y − x ) cos ⁡ ( y ) . {\displaystyle \sin(y)-\sin(x)\approx (y-x)\cos(y).} This can be interpreted as

11433-566: 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

11544-514: Was a treatise, the origin being Kepler's methods, written by Bonaventura Cavalieri , who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. The ideas were similar to Archimedes' in The Method , but this treatise is believed to have been lost in the 13th century and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. Cavalieri's work

11655-461: Was achieved by John Wallis , Isaac Barrow , and James Gregory , the latter two proving predecessors to the second fundamental theorem of calculus around 1670. The product rule and chain rule , the notions of higher derivatives and Taylor series , and of analytic functions were used by Isaac Newton in an idiosyncratic notation which he applied to solve problems of mathematical physics . In his works, Newton rephrased his ideas to suit

11766-488: Was clear that he understood the principles of the Taylor series . He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable. These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz , who was originally accused of plagiarism by Newton. He is now regarded as an independent inventor of and contributor to calculus. His contribution

11877-403: Was formulated separately in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz . Later work, including codifying the idea of limits , put these developments on a more solid conceptual footing. Today, calculus is widely used in science , engineering , biology , and even has applications in social science and other branches of math. In mathematics education , calculus

11988-494: Was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. Pierre de Fermat , claiming that he borrowed from Diophantus , introduced the concept of adequality , which represented equality up to an infinitesimal error term. The combination

12099-445: Was the first to apply calculus to general physics . Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, emphasizing that differentiation and integration are inverse processes, second and higher derivatives, and the notion of an approximating polynomial series. When Newton and Leibniz first published their results, there

12210-419: Was their ratio. The infinitesimal approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. In the late 19th century, infinitesimals were replaced within academia by the epsilon, delta approach to limits . Limits describe the behavior of a function at a certain input in terms of its values at nearby inputs. They capture small-scale behavior using

12321-425: 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. Today, Leibniz and Newton are usually both given credit for independently inventing and developing calculus. Newton

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