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In mathematics , differential calculus is a subfield of calculus  concerned with the study of the rates at which quantities change.
It is one of the two traditional divisions of calculus, the other being integral calculus , the study of the area beneath a curve.
The primary objects of study in differential calculus are the derivative of a function , related notions such as the differential , and their applications.
The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point.
For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point. Differential calculus and integral calculus are connected by the fundamental theorem of calculus , which states that differentiation is the reverse process to integration.
Differentiation has applications to nearly all quantitative disciplines. For example, 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.
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. Derivatives are frequently used to find the maxima and minima of a function.
Patented Library of Calculus Spreadsheet Functions
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.
Suppose that x and y are real numbers and that y is a function of x , that is, for every value of x , there is a corresponding value of y. In this "slope-intercept form", the term m is called the slope and can be determined from the formula:.
A general function is not a line, so it does not have a slope. Since the derivative is the slope of the linear approximation to f at the point a , the derivative together with the value of f at a determines the best linear approximation, or linearization , of f near the point a.
If every point a in the domain of f has a derivative, there is a function that sends every point a to the derivative of f at a. A closely related notion is the differential of a function. When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph of f at x. Because the source and target of f are one-dimensional, the derivative of f is a real number.
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If x and y are vectors, then the best linear approximation to the graph of f depends on how f changes in several directions at once. The linearization of f in all directions at once is called the total derivative. 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. The use of infinitesimals to study rates of change can be found in Indian mathematics , perhaps as early as AD, when the astronomer and mathematician Aryabhata — used infinitesimals to study the orbit of the Moon.
Rashed's conclusion has been contested by other scholars, however, who argue that he could have obtained the result by other methods which do not require the derivative of the function to be known. The modern development of calculus is usually credited to Isaac Newton — and Gottfried Wilhelm Leibniz — , 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. Regarding Fermat's influence, Newton once wrote in a letter that " I had the hint of this method [of fluxions] from Fermat's way of drawing tangents, and by applying it to abstract equations, directly and invertedly, I made it general.
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 — , Bernhard Riemann — , and Karl Weierstrass — It was also during this period that the differentiation was generalized to Euclidean space and the complex plane.
If f is not assumed to be everywhere differentiable, then points at which it fails to be differentiable are also designated critical points.
This is called the second derivative test. An alternative approach, called the first derivative test , involves considering the sign of the f' on each side of the critical point.
Taking derivatives and solving for critical points is therefore often a simple way to find local minima or maxima, which can be useful in optimization. By the extreme value theorem , a continuous function on a closed interval must attain its minimum and maximum values at least once.
If the function is differentiable, the minima and maxima can only occur at critical points or endpoints.
Implicit Differentiation Explained - Product Rule, Quotient & Chain Rule - Calculus
This also has applications in graph sketching: once the local minima and maxima of a differentiable function have been found, a rough plot of the graph can be obtained from the observation that it will be either increasing or decreasing between critical points.
In higher dimensions , a critical point of a scalar valued function is a point at which the gradient is zero. The second derivative test can still be used to analyse critical points by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point.
If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is called a " saddle point ", and if none of these cases hold i.
Differential calculus derivatives pdf to excel
One example of an optimization problem is: Find the shortest curve between two points on a surface, assuming that the curve must also lie on the surface. If the surface is a plane, then the shortest curve is a line. But if the surface is, for example, egg-shaped, then the shortest path is not immediately clear.
These paths are called geodesics , and one of the most fundamental problems in the calculus of variations is finding geodesics. Another example is: Find the smallest area surface filling in a closed curve in space. This surface is called a minimal surface and it, too, can be found using the calculus of variations. Calculus is of vital importance in physics: many physical processes are described by equations involving derivatives, called differential equations.
Physics is particularly concerned with the way quantities change and develop over time, and the concept of the " time derivative " — the rate of change over time — is essential for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics :. A differential equation is a relation between a collection of functions and their derivatives. An ordinary differential equation is a differential equation that relates functions of one variable to their derivatives with respect to that variable.
A partial differential equation is a differential equation that relates functions of more than one variable to their partial derivatives. Differential equations arise naturally in the physical sciences, in mathematical modelling, and within mathematics itself. For example, Newton's second law , which describes the relationship between acceleration and force, can be stated as the ordinary differential equation.
The heat equation in one space variable, which describes how heat diffuses through a straight rod, is the partial differential equation. The mean value theorem gives a relationship between values of the derivative and values of the original function. In other words,. In practice, what the mean value theorem does is control a function in terms of its derivative.
Spreadsheet Calculus: Derivatives and Integrals
For instance, suppose that f has derivative equal to zero at each point. This means that its tangent line is horizontal at every point, so the function should also be horizontal. The mean value theorem proves that this must be true: The slope between any two points on the graph of f must equal the slope of one of the tangent lines of f. All of those slopes are zero, so any line from one point on the graph to another point will also have slope zero.
But that says that the function does not move up or down, so it must be a horizontal line. More complicated conditions on the derivative lead to less precise but still highly useful information about the original function.
The derivative gives the best possible linear approximation of a function at a given point, but this can be very different from the original function.
Calculate a Derivative in Excel from Tables of Data
One way of improving the approximation is to take a quadratic approximation. For each one of these polynomials, there should be a best possible choice of coefficients a , b , c , and d that makes the approximation as good as possible. In the neighbourhood of x 0 , for a the best possible choice is always f x 0 , and for b the best possible choice is always f' x 0. For c , d , and higher-degree coefficients, these coefficients are determined by higher derivatives of f.
Using these coefficients gives the Taylor polynomial of f. The Taylor polynomial of degree d is the polynomial of degree d which best approximates f , and its coefficients can be found by a generalization of the above formulas. Taylor's theorem gives a precise bound on how good the approximation is. If f is a polynomial of degree less than or equal to d , then the Taylor polynomial of degree d equals f.
The limit of the Taylor polynomials is an infinite series called the Taylor series. The Taylor series is frequently a very good approximation to the original function. Functions which are equal to their Taylor series are called analytic functions. It is impossible for functions with discontinuities or sharp corners to be analytic, but there are smooth functions which are not analytic.
Some natural geometric shapes, such as circles , cannot be drawn as the graph of a function. This set is called the zero set of f , and is not the same as the graph of f , which is a paraboloid. It states that if f is continuously differentiable , then around most points, the zero set of f looks like graphs of functions pasted together.
The points where this is not true are determined by a condition on the derivative of f. The implicit function theorem is closely related to the inverse function theorem , which states when a function looks like graphs of invertible functions pasted together. From Wikipedia, the free encyclopedia.
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Fractional Malliavin Stochastic Variations. Glossary of calculus. Main article: Derivative.
Main article: History of calculus. Main article: Calculus of variations. Main article: Differential equation. Main article: Mean value theorem.