Concept of slope

1. Concept of slope Slope in geometry, algebra and analysis, And logarithmic derivative.

2. Slope The slope of a line is defined as the rise over the run, m = ?y / ?x. Slope is used to describe the steepness, incline, gradient, or grade of a straight line. A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. It is also always the same thing as how many rises in one run. Using calculus, one can calculate the slope of the tangent to a curve at a point.

3. Definition The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. This is described by the following equation: (The delta symbol, "?", is commonly used in mathematics to mean "difference" or "change".) Given two points (x1, y1) and (x2, y2), the change in x from one to the other is x2 - x1, while the change in y is y2 - y1. Substituting both quantities into the above equation obtains the following: Note that the way the points are chosen on the line and their order does not matter; the slope will be the same in each case. Other curves have "accelerating" slopes and one can use calculus to determine such slopes.

4. Geometry The larger the absolute value of a slope, the steeper the line. A horizontal line has slope 0, a 45� rising line has a slope of +1, and a 45� falling line has a slope of -1. A vertical line's slope is undefined meaning it has "no slope." The angle ? a line makes with the positive x axis is closely related to the slope m via the tangent function: and Two lines are parallel if and only if their slopes are equal and they are not coincident or if they both are vertical and therefore have undefined slopes. Two lines are perpendicular if the product of their slopes is -1 or one has a slope of 0 (a horizontal line) and the other has an undefined slope (a vertical line). Also, another way to determine a perpendicular line is to find the slope of one line and then to get its reciprocal and then reversing its positive or negative sign (e.g. a line perpendicular to a line of slope -2 is +1/2).

5. Slope of a road or railway There are two common ways to describe how steep a road or railroad is. One is by the angle in degrees, and the other is by the slope in a percentage. See also mountain railway and rack railway. The formulae for converting a slope as a percentage into an angle in degrees and vice versa are: and where angle is in degrees and the trigonometry functions operate in degrees. For example, a 100% or 1000� slope is 45�. A third way is to give one unit of rise in say 10, 20, 50 or 100 horizontal units, e.g. 1:10. 1:20, 1:50 or 1:100 (etc.). .

6. Algebra If y is a linear function of x, then the coefficient of x is the slope of the line created by plotting the function. Therefore, if the equation of the line is given in the form then m is the slope. This form of a line's equation is called the slope-intercept form, because b can be interpreted as the y-intercept of the line, the y-coordinate where the line intersects the y-axis. If the slope m of a line and a point (x0, y0) on the line are both known, then the equation of the line can be found using the point-slope formula:

7. Calculus The concept of a slope is central to differential calculus. For non-linear functions, the rate of change varies along the curve. The derivative of the function at a point is the slope of the line tangent to the curve at the point, and is thus equal to the rate of change of the function at that point. If we let ?x and ?y be the distances (along the x and y axes, respectively) between two points on a curve, then the slope given by the above definition, is the slope of a secant line to the curve. For a line, the secant between any two points is the line itself, but this is not the case for any other type of curve.

8. An example At each point, the derivative is the slope of a line that is tangent to the curve. The line is always tangent to the blue curve; its slope is the derivative. Note derivative is positive where green, negative where red, and zero where black By moving the two points closer together so that ?y and ?x decrease, the secant line more closely approximates a tangent line to the curve, and as such the slope of the secant approaches that of the tangent. Using differential calculus, we can determine the limit, or the value that ?y/?x approaches as ?y and ?x get closer to zero; it follows that this limit is the exact slope of the tangent. If y is dependent on x, then it is sufficient to take the limit where only ?x approaches zero. Therefore, the slope of the tangent is the limit of ?y/?x as ?x approaches zero. We call this limit the derivative

9. Related rates In differential calculus, related rates problems involve finding a rate that a quantity changes by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time. The most common way to approach related rates problems is the following: Identify the known rates of change and the rate of change that is to be found. Construct an equation relating the quantities whose rates of change are known to the quantity whose rate of change is to be found. Differentiate both sides of the equation with respect to time (or other rate of change). Substitute the known rates of change and the known quantities into the equation. Solve for the wanted rate of change.

10. Method of Fluxions Method of Fluxions is a book by Isaac Newton. The book was completed in 1671, and published in 1736. Fluxions is Newton's term for differential calculus (fluents was his term for integral calculus). He originally developed the method at Woolsthorpe Manor during the closing of Cambridge during the Great Plague of London from 1665 to 1667, but did not choose to make his findings known (similarly, his findings which eventually became the Philosophiae Naturalis Principia Mathematica were developed at this time and hidden from the world in Newton's notes for many years). Gottfried Leibniz developed his calculus around 1673, and published it in 1684, twenty years before Newton. The calculus notation we use today is mostly that of Leibniz, although Newton's dot notation for differentiation for denoting derivatives with respect to time is still in current use throughout mechanics. Newton's Method of Fluxions was formally published posthumously, but following Leibniz's publication of the calculus a bitter rivalry erupted between the two mathematicians over who had developed the calculus first and so Newton no longer hid his knowledge of fluxions.

11. Tangent In geometry, the tangent line (or simply the tangent) to a curve at a given point is the straight line that "just touches" the curve at that point (in the sense explained more precisely below). As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point. The same definition applies to space curves and curves in n-dimensional Euclidean space. Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space. In trigonometry, tangent is the slope formula that combines the rise and run. The tangent function takes an angle and tells the slope when the length of the line is 1. For more information on the tangent function, see trigonometric functions. The word "tangent" comes from the Latin tangere, meaning "to touch".

12. The intuitive notion that a tangent line "just touches" a curve can be made more explicit by considering the sequence of straight lines (secant lines) passing through two points, A and B, those that lie on the function curve. The tangent at A is the limit when point B approximates or tends to A. The existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as "differentiability". For example, if two circular arcs meet at a sharp point (a vertex) then there is no uniquely defined tangent at the vertex because the limit of the progression of secant lines depends on the direction in which "point B" approaches the vertex. In most cases commonly encountered, the tangent to a curve does not cross the curve at the point of tangency (though it may, when continued, cross the curve at other places away from the point of tangent) This is true, for example, of all tangents to a circle or a parabola. However, at exceptional points called inflection points, the tangent line does cross the curve at the point of tangency. An example is the point (0,0) on the graph of the cubic parabola y = x3. Conversely, it may happen that the curve lies entirely on one side of a straight line passing through a point on it, and yet this straight line is not a tangent line. This is the case, for example, for a line passing through the vertex of a triangle and not intersecting the triangle�where the tangent line does not exist for the reasons explained above. In convex geometry, such lines are called supporting lines.

13. Intuitive description Suppose that a curve is given as the graph of a function, y = f(x). To find the tangent line at the point p = (a, f(a)), consider another nearby point q = (a + h, f(a + h)) on the curve. The slope of the secant line passing through p and q is equal to the difference quotient As the point q approaches p, which corresponds to making h smaller and smaller, the difference quotient should approach a certain limiting value k, which is the slope of the tangent line at the point p. If k is known, the equation of the tangent line can be found in the point-slope form:

14. More rigorous description To make the preceding reasoning rigorous, one has to explain what is meant by the difference quotient approaching a certain limiting value k. The precise mathematical formulation was given by Cauchy in the 19th century and is based on the notion of limit. Suppose that the graph does not have a break or a sharp edge at p and it is neither plumb nor too wiggly near p. Then there is a unique value of k such that as h approaches 0, the difference quotient gets closer and closer to k, and the distance between them becomes negligible compared with the size of h, if h is small enough. This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function f. This limit is the derivative of the function f at x = a, denoted f�'(a). Using derivatives, the equation of the tangent line can be stated as follows: Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the power function, trigonometric functions, exponential function, logarithm, and their various combinations. Thus, equations of the tangents to graphs of all these functions, as well as many others, can be found by the methods of calculus.

15. When the method fails Calculus also demonstrates that there are functions and points on their graphs for which the limit determining the slope of the tangent line does not exist. For these points the function f is non-differentiable. There are two possible reasons for the method of finding the tangents based on the limits and derivatives to fail: either the geometric tangent exists, but it is a vertical line, which cannot be given in the point-slope form since it does not have a slope, or the graph is too badly behaved to admit a geometric tangent. The graph y = x1/3 illustrates the first possibility: here the difference quotient at a = 0 is equal to h1/3/h = h-2/3, which becomes very large as h approaches 0. The tangent line to this curve at the origin is vertical. The graph y = |x| of the absolute value function consists of two straight lines with different slopes joined at the origin. As a point q approaches the origin from the right, the secant line always has slope 1. As a point q approaches the origin from the left, the secant line always has slope -1. Therefore, there is no unique tangent to the graph at the origin (although in a certain sense, there are two half-tangents, corresponding to two possible directions of approaching the origin).

16. Differential calculus Differential calculus, a field in mathematics, is the study of how functions change when their inputs change. The primary object of study in differential calculus is the derivative. A closely related notion is the differential. The derivative of a function at a chosen input value describes the behavior of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In general, the derivative of a function at a point determines the best linear approximation to the function at that point. The process of finding a derivative is called differentiation. The fundamental theorem of calculus states that differentiation is the reverse process to integration.

17. Applications of derivation Differentiation has applications to all quantitative disciplines. In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. Newton's second law of motion states that the derivative of the momentum of a body equals the force applied to the body. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories. By applying game theory, differentiation can provide best strategies for competing corporations. Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory and abstract algebra.

18. The derivative Suppose that x and y are real numbers and that y is a function of x, that is, y = f(x). One of the simplest types of functions is a linear function. This is a function whose graph is a line. In this case, y = f(x) = m x + c, where m and c are real numbers that depend on which line the graph determines. m is called the slope and is given by where the symbol ? (the uppercase form of the Greek letter Delta) is an abbreviation for "change in". This formula is true because y + ?y = f(x + ?x) = m (x + ?x) + c = m x + c + m ?x = y + m?x. It follows that ?y = m ?x. In linear functions the derivative of f at the point x is the best possible approximation to the idea of the slope of f at the point x. It is usually denoted f'(x) or dy/dx. Together with the value of f at x, the derivative of f determines the best linear approximation, or linearization, of f near the point x. This latter property is usually taken as the definition of the derivative. Derivatives cannot be calculated in nonlinear functions because they do not have a well-defined slope.

19. History of differentiation The concept of a derivative in the sense of a tangent line is a very old one, familiar to Greek geometers such as Euclid (c. 300 BCE), Archimedes (c. 287�212 BCE) and Apollonius of Perga (c. 262�190 BCE). Archimedes also introduced the use of infinitesimals, although these were primarily used to study areas and volumes rather than derivatives and tangents; see Archimedes' use of infinitesimals. The use of infinitesimals to study rates of change can be found in Indian mathematics, perhaps as early as 500 CE, when the astronomer and mathematician Aryabhata (476�550) used infinitesimals to study the motion of the moon. The use of infinitesimals to compute rates of change was developed significantly by Bhaskara II (1114-1185); indeed, it has been argued that many of the key notions of differential calculus can be found in his work, such as "Rolle's theorem". The Persian mathematician, Sharaf al-Din al-Tusi (1135-1213), was the first to discover the derivative of cubic polynomials, an important result in differential calculus; his Treatise on Equations developed concepts related to differential calculus, such as the derivative function and the maxima and minima of curves, in order to solve cubic equations which may not have positive solutions. An early version of the mean value theorem was first described by Parameshvara (1370�1460) from the Kerala school of astronomy and mathematics in his commentary on Bhaskara II.

20. The modern development of calculus is usually credited to Isaac Newton (1643 � 1727) and Gottfried Leibniz (1646 � 1716), who provided independent and unified approaches to differentiation and derivatives. The key insight, however, that earned them this credit, was the fundamental theorem of calculus relating differentiation and integration: this rendered obsolete most previous methods for computing areas and volumes, which had not been significantly extended since the time of Ibn al-Haytham (Alhazen). For their ideas on derivatives, both Newton and Leibniz built on significant earlier work by mathematicians such as Isaac Barrow (1630 � 1677), Ren� Descartes (1596 � 1650), Christiaan Huygens (1629 � 1695), Blaise Pascal (1623 � 1662) and John Wallis (1616 � 1703). In particular, Isaac Barrow is often credited with the early development of the derivative. evertheless, Newton and Leibniz remain key figures in the history of differentiation, not least because Newton was the first to apply differentiation to theoretical physics, while Leibniz systematically developed much of the notation still used today. Since the 17th century many mathematicians have contributed to the theory of differentiation. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Augustin Louis Cauchy (1789 � 1857), Bernhard Riemann (1826 � 1866), and Karl Weierstrass (1815 � 1897). It was also during this period that the differentiation was generalized to Euclidean space and the complex plane.

21. Power rule The power rule for differentiation states that for every natural number n To prove the power rule for differentiation, we use the definition of the derivative as a limit: Substituting f(x) = xn gives One can then express (x + h)n by applying the binomial theorem to obtain The i = n term of the sum can then be written independently of the sum to yield Canceling the xn terms one generates

22. Conclusion of proof An h can be factored out from each term in the sum to give From thence we can cancel the h in the denominator to obtain To evaluate this limit we observe that n - i - 1 > 0 for all i < n - 1 and equal to zero for i = n - 1. Thus only the h0 term will survive with i = n - 1 yielding Evaluating the binomial coefficient gives It follows that

23. Logarithmic derivative In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula where f ' is the derivative of f. When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is equal to ln(f); or, the derivative of the natural logarithm of f. This follows directly from the chain rule. Both derivatives and logarithms have a product rule, a reciprocal rule, a quotient rule, and a power rule (compare the list of logarithmic identities); each pair of rules is related through the logarithmic derivative.

24. Basic properties Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does not take values in the positive reals. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have So for positive-real-valued functions, the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors. But we can also use the Leibniz law for the derivative of a product to get Thus, it's true for any function that the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors (when they are defined).

25. Similarly (in fact this is a consequence), the logarithmic derivative of the reciprocal of a function is the negation of the logarithmic derivative of the function: just as the logarithm of the reciprocal of a positive real number is the negation of the logarithm of the number. More generally, the logarithmic derivative of a quotient is the difference of the logarithmic derivatives of the dividend and the divisor: just as the logarithm of a quotient is the difference of the logarithms of the dividend and the divisor. Generalising in another direction, the logarithmic derivative of a power (with constant real exponent) is the product of the exponent and the logarithmic derivative of the base: just as the logarithm of a power is the product of the exponent and the logarithm of the base.

26. Computing ordinary derivatives using logarithmic derivatives Logarithmic derivatives can simplify the computation of derivatives requiring the product rule. The procedure is as follows: Suppose that �(x)�=�u(x)v(x) and that we wish to compute �'(x). Instead of computing it directly, we compute its logarithmic derivative. That is, we compute: Multiplying through by � computes �': This technique is most useful when � is a product of a large number of factors. This technique makes it possible to compute �' by computing the logarithmic derivative of each factor, summing, and multiplying by �.

27. Derivatives of simple functions .

28. Derivatives of exponential and logarithmic functions note that the equation above is true for all c, but the derivative yields a complex number.

29. Derivatives of trigonometric functions

30. Derivatives of hyperbolic functions .