We usually write differentials as dx, dy, dt (and so on), where: dx is an infinitely small change in x; dy is an infinitely small change in y; and. Page 1 of 25 DIFFERENTIATION II In this article we shall investigate some mathematical applications of differentiation. The first-order logic of this new set of hyperreal numbers is the same as the logic for the usual real numbers, but the completeness axiom (which involves second-order logic) does not hold. y = f(x) is written: Note: We are now treating dy/dx more like a fraction (where we can manipulate the parts separately), rather than as an operator. Do you believe the recommendations are re }dt(and so on), where: When comparing small changes in quantities that are related to each other (like in the case where y\displaystyle{y}y is some function f x\displaystyle{x}x, we say the differential dy\displaystyle{\left.{d}{y}\right. Tailor assignments based on students’ learning goals – Using differentiation strategies to shake up … These approaches are very different from each other, but they have in common the idea of being quantitative, i.e., saying not just that a differential is infinitely small, but how small it is. Hence, if f is differentiable on all of Rn, we can write, more concisely: This idea generalizes straightforwardly to functions from Rn to Rm. Differentials are infinitely small quantities. A small change in radius will be multiplied by 125.7, whereas a small change in height will be multiplied by 12.57. Consider a function $$f$$ that is differentiable at point $$a$$. Google uses integration to speed up the Web, Factoring trig equations (2) by phinah [Solved! What did it say? Furthermore, it has the decisive advantage over other definitions of the derivative that it is invariant under changes of coordinates. If δx is very small, δy δx will be a good approximation of dy dx, If δ x is very small, δ y δ x will be a good approximation of d y d x,, This is very useful information in determining an approximation of the change in one variable given the small change in the second variable. To illustrate, suppose f(x) is a real-valued function on R. We can reinterpret the variable x in f(x) as being a function rather than a number, namely the identity map on the real line, which takes a real number p to itself: x(p) = p. Then f(x) is the composite of f with x, whose value at p is f(x(p)) = f(p). ... We examine change for differentiation at the school level rather than at the individual teacher or district level. The main idea of this approach is to replace the category of sets with another category of smoothly varying sets which is a topos. The idea of an infinitely small or infinitely slow change is, intuitively, extremely useful, and there are a number of ways to make the notion mathematically precise. Section 4-1 : Rates of Change. In this video, you will learn two different type of small change questions, to help u fully understand about the small change topic. The only precise way of defining f (x) in terms of f' (x) is by evaluating f' (x) Δx over infinitely small intervals, keeping in mind that f. The point and the point P are joined in a line that is the tangent of the curve. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.. Differentials are infinitely small quantities. The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. },dx, dy,\displaystyle{\left.{d}{y}\right. A series of rules have been derived for differentiating various types of functions. In Leibniz's notation, if x is a variable quantity, then dx denotes an infinitesimal change in the variable x. Consider a function defined by y = f(x). 4 Differentiation. We learned that the derivative or rate of change of a function can be written as , where dy is an infinitely small change in y, and dx (or \Delta x) is an infinitely small change in x. Thus we recover the idea that f ′ is the ratio of the differentials df and dx. Thus, if y is a function of x, then the derivative of y with respect to x is often denoted dy/dx, which would otherwise be denoted (in the notation of Newton or Lagrange) ẏ or y′. Use differentiation to find the small change in y when x increases from 2 to 2.02. the integral sign (which is a modified long s) denotes the infinite sum, f(x) denotes the "height" of a thin strip, and the differential dx denotes its infinitely thin width. This is an application that we repeatedly saw in the previous chapter. Let us discuss the important terms involved in the differential calculus basics. That is, The differential of the dependent variable y, … ], Different parabola equation when finding area by phinah [Solved!]. After all, we can very easily compute $$f(4.1,0.8)$$ using readily available technology. In algebraic geometry, differentials and other infinitesimal notions are handled in a very explicit way by accepting that the coordinate ring or structure sheaf of a space may contain nilpotent elements. To express the rate of change in any function we introduce concept of derivative which involves a very small change in the dependent variable with reference to a very small change in independent variable. the notation used in integration. This formula summarizes the intuitive idea that the derivative of y with respect to x is the limit of the ratio of differences Δy/Δx as Δx becomes infinitesimal. Find the differential dy of the function y = 5x^2-4x+2. Suppose the input $$x$$ changes by a small amount. }dy, o… Such a thickened point is a simple example of a scheme.. lim_(Delta x->0) (Delta y)/(Delta x)=dy/dx. The point of the previous example was not to develop an approximation method for known functions. Complete and updated to the latest syllabus. Sometimes you will find this in science textbooks as well for small changes, but it should be avoided. Differentiation is the process of finding a derivative. The simplest example is the ring of dual numbers R[ε], where ε2 = 0. For other uses of "differential" in mathematics, see, https://en.wikipedia.org/w/index.php?title=Differential_(infinitesimal)&oldid=979585401, All articles with specifically marked weasel-worded phrases, Articles with specifically marked weasel-worded phrases from November 2012, Creative Commons Attribution-ShareAlike License, Differentials in smooth models of set theory. It turns out that if f\left( x \right) is a function that is differentiable on an open interval containing x, and the differential of x (dx) is a non-zero real number, then dy={f}’\left( x \right)dx (see how we just multiplied b… Privacy & Cookies | We used d/dx as an operator. Small changes are easier to make, and chances are those changes will stick with you and become part of your habits. When the point Q is move nearer and neared to the point P, there will be a point which is very near to point P but not the point P and there is a very small change in value of x and y at the point from point P. 2. do this, but it is pretty silly, since we can easily find the exact change - why approximate it? DN1.11: SMALL CHANGES AND . We learned before in the Differentiation chapter that the slope of a curve at point P is given by dy/dx., Relationship between dx, dy, Delta x, and Delta y. Functions. However the logic in this new category is not identical to the familiar logic of the category of sets: in particular, the law of the excluded middle does not hold. Example 1 Given that y = 3x 2+ 2x -4. The derivative of a function is the rate of change of the output value with respect to its input value, whereas differential is the actual change of function. where, assuming h and k to be small, we have ignored higher-order terms involving powers of h and k. We deﬁne δf to be the change in f(x,y) resulting from small changes to x 0 and y 0, denoted by h and k respectively. Our advice is to take small steps. The term differential is used in calculus to refer to an infinitesimal (infinitely small) change in some varying quantity. It identifies … The differential dfp has the same property, because it is just a multiple of dxp, and this multiple is the derivative f ′(p) by definition. If x is increased by a small amount ∆x to x + ∆ x, then as ∆ x → 0, y x ∆ ∆ → dy dx. The differential dx represents an infinitely small change in the variable x. For example, if x is a variable, then a change in the value of x is often denoted Δx (pronounced delta x). Take time to re ect on the recommendations. We now connect differentials to linear approximations. Although it is an aim of differentiation to focus on individuals, it is not a goal to make individual lesson plans for each student. We could use the differential to estimate the $\frac{d}{dx}$ Used to represent derivatives and integrals.  Isaac Newton referred to them as fluxions. Algebraic geometers regard this equivalence class as the restriction of f to a thickened version of the point p whose coordinate ring is not R (which is the quotient space of functions on R modulo Ip) but R[ε] which is the quotient space of functions on R modulo Ip2. The previous example showed that the volume of a particular tank was more sensitive to changes in radius than in height. This would just be a trick were it not for the fact that: For instance, if f is a function from Rn to R, then we say that f is differentiable at p ∈ Rn if there is a linear map dfp from Rn to R such that for any ε > 0, there is a neighbourhood N of p such that for x ∈ N. We can now use the same trick as in the one-dimensional case and think of the expression f(x1, x2, ..., xn) as the composite of f with the standard coordinates x1, x2, ..., xn on Rn (so that xj(p) is the j-th component of p ∈ Rn). Differentials can be used to estimate the change in the value of a function resulting from a small change in input values. Differentiation is a process where we find the derivative of a function. 2. [ δy = 0.28 ]  Such extensions of the real numbers may be constructed explicitly using equivalence classes of sequences of real numbers, so that, for example, the sequence (1, 1/2, 1/3, ..., 1/n, ...) represents an infinitesimal. A third approach to infinitesimals is the method of synthetic differential geometry or smooth infinitesimal analysis. Use $\delta$ instead. real change in value of a function (Δy) caused by a small We now go on to see how the differential is used to perform the opposite process of differentiation, which first we'll call antidifferentiation, and later integration. Thus: δf = f(x 0 +h, y 0 +k)−f(x 0,y 0) and so δf ’ hf x(x 0,y 0) + kf y(x 0,y 0). About & Contact | In this video I go through how to solve an equation using the method of small increments. Thus the volume of the tank is more sensitive to changes in radius than in height. This ratio holds true even when the changes approach zero. Earlier in the differentiation chapter, we wrote dy/dx and f'(x) to mean the same thing. 5.1 Reverse to differentiation; 5.2 What is constant of integration? The differential dx represents an infinitely small change in the variable x. Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, dy being the infinitesimally small change in y caused by an infinitesimally small change dx applied to x. the impact of a unit change in x … What did Isaac Newton's original manuscript look like? Think of differentials of picking apart the “fraction” \displaystyle \frac{{dy}}{{dx}} we learned to use when differentiating a function. The identity map has the property that if ε is very small, then dxp(ε) is very small, which enables us to regard it as infinitesimal. We usually write differentials as dx,\displaystyle{\left.{d}{x}\right. Infinitesimal quantities played a significant role in the development of calculus. This value is the same at any point on a straight- line graph. The differential df (which of course depends on f) is then a function whose value at p (usually denoted dfp) is not a number, but a linear map from R to R. Since a linear map from R to R is given by a 1×1 matrix, it is essentially the same thing as a number, but the change in the point of view allows us to think of dfp as an infinitesimal and compare it with the standard infinitesimal dxp, which is again just the identity map from R to R (a 1×1 matrix with entry 1). Home | The term differential is used in calculus to refer to an infinitesimal (infinitely small) change in some varying quantity. Many text books As stated above, derivative of a function represents the change in the dependent variable due to a infinitesimally small change in the independent variable and is written as dY / dX for a function Y = f (X). This week's Friday Math Movie is an explanation of differentials, a calculus topic. See Slope of a tangent for some background on this. The partial-derivative relations derived in Problems 1, 4, and 5, plus a bit more partial-derivative trickery, can be used to derive a completely general relation between C p and C V. (a) With the heat capacity expressions from Problem 4 in mind, first consider S to be a function of T and V.Expand dS in terms of the partial derivatives (∂ S / ∂ T) V and (∂ S / ∂ V) T. That is, The differential of the independent variable x is written dx and is the same as the change in x, Δ x. When the point Q is move nearer and neared to the point P, there will be a point which is very near to point P but not the point P and there is a very small change in value of x and y at the point from point P. 2. There are several approaches for making the notion of differentials mathematically precise. However it is not a sufficient condition. DN1.11 – Differentiation:: : Small Changes and Approximations Page 1 of 3 June 2012. However, it was Gottfried Leibniz who coined the term differentials for infinitesimal quantities and introduced the notation for them which is still used today. There is a simple way to make precise sense of differentials by regarding them as linear maps. We are introducing differentials here as an introduction to Solve your calculus problem step by step! Nevertheless, this suffices to develop an elementary and quite intuitive approach to calculus using infinitesimals, see transfer principle. This means that the same idea can be used to define the differential of smooth maps between smooth manifolds. This video will teach you how to determine their term (dy/dt or dy/dx or dx/dt) by using the units given by the question. In this category, one can define the real numbers, smooth functions, and so on, but the real numbers automatically contain nilpotent infinitesimals, so these do not need to be introduced by hand as in the algebraic geometric approach. APPROXIMATIONS . The symbol d is used to denote a change that is infinitesimally small. That is the fact that $$f'\left( x \right)$$ represents the rate of change of $$f\left( x \right)$$. dt is an infinitely small change in t. We will use this new form of the derivative throughout this chapter on Integration. In the nonstandard analysis approach there are no nilpotent infinitesimals, only invertible ones, which may be viewed as the reciprocals of infinitely large numbers. reading the recommendations. This calculus solver can solve a wide range of math problems. Derivative or Differentiation of a function For a small change in variable x x, the rate of change in the function f (x) f … Free CAIE IGCSE Add Maths (0606) Theory Differentiation & Integration summarized revision notes written for students, by students. This can be motivated by the algebro-geometric point of view on the derivative of a function f from R to R at a point p. For this, note first that f − f(p) belongs to the ideal Ip of functions on R which vanish at p. If the derivative f vanishes at p, then f − f(p) belongs to the square Ip2 of this ideal. approximation of the change in one variable given the small change in the second variable. Aside: Note that the existence of all the partial derivatives of f(x) at x is a necessary condition for the existence of a differential at x. The slope of the dashed line is given by the ratio (Delta y)/(Delta x). As Delta x gets smaller, that slope becomes closer to the actual slope at P, which is the "instantaneous" ratio dy/dx. The change in the function is only valid for the derivative evaluated at a point multiplied by an infinitely small dx The derivative is only constant over an infinitely small interval,. On our graph the ratios are all the same and equal to the velocity. Focused on individuals, small groups, and the class as a whole. In this page, differentiation is defined in first principles : instantaneous rate of change is the change in a quantity for a small change δ → 0 δ → 0 in the variable. Then the differentials (dx1)p, (dx2)p, (dxn)p at a point p form a basis for the vector space of linear maps from Rn to R and therefore, if f is differentiable at p, we can write dfp as a linear combination of these basis elements: The coefficients Djf(p) are (by definition) the partial derivatives of f at p with respect to x1, x2, ..., xn. v = dx/dt =x/t = x/t. regard this disadvantage as a positive thing, since it forces one to find constructive arguments wherever they are available. The point and the point P are joined in a line that is the tangent of the curve. Author: Murray Bourne | To find the differential dy, we just need to find the derivative and write it with dx on the right. and . Some[who?] What did Newton originally say about Integration? In an expression such as. We now see a different way to write, and to think about, the derivative. This approach is known as, it captures the idea of the derivative of, This page was last edited on 21 September 2020, at 15:29. change in x (written as Δx). I hope it helps :) If y is a function of x, then the differential dy of y is related to dx by the formula. Antiderivatives and The Indefinite Integral, Different parabola equation when finding area. Look at the people in your life you respect and admire for their accomplishments. Find the differential dy of the function y = 3x^5- x. We therefore obtain that dfp = f ′(p) dxp, and hence df = f ′ dx. 4.1 Rate of change; 4.2 Average rate of change across an interval; 4.3 Rate of change at a point; 4.4 Terminology and notation; 4.5 Table of derivatives; 4.6 Exercises (differentiation) Answers to selected exercises (differentiation) 5 Integration. y x ∆ ∆ ≈ dy dx. IntMath feed |. Applications of Differentiation . If Δ x is very small (Δ x ≠ 0), then the slope of the tangent is approximately the same as the slope of the secant line through ( x, f(x)). Measuring change in a linear function: y = a + bx a = intercept b = constant slope i.e. This means that set-theoretic mathematical arguments only extend to smooth infinitesimal analysis if they are constructive (e.g., do not use proof by contradiction). The final approach to infinitesimals again involves extending the real numbers, but in a less drastic way.  This is closely related to the algebraic-geometric approach, except that the infinitesimals are more implicit and intuitive. Nevertheless, the notation has remained popular because it suggests strongly the idea that the derivative of y at x is its instantaneous rate of change (the slope of the graph's tangent line), which may be obtained by taking the limit of the ratio Δy/Δx of the change in y over the change in x, as the change in x becomes arbitrarily small. The use of differentials in this form attracted much criticism, for instance in the famous pamphlet The Analyst by Bishop Berkeley. where dy/dx denotes the derivative of y with respect to x. ... To find the approximate value of small change in a quantity; Real-life applications of differential calculus are: Sitemap | Free CAIE IGCSE Add Maths (0606) Theory Differentiation & Integration summarized revision notes written for students, by students. },dy, dt\displaystyle{\left.{d}{t}\right. Delta y means "change in y, and Delta x means "change in x". Differentials are also compatible with dimensional analysis, where a differential such as dx has the same dimensions as the variable x. Differentials are also used in the notation for integrals because an integral can be regarded as an infinite sum of infinitesimal quantities: the area under a graph is obtained by subdividing the graph into infinitely thin strips and summing their areas. Product differentiation is intended to prod the consumer into choosing one brand over another in a crowded field of competitors. In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. Hence the derivative of f at p may be captured by the equivalence class [f − f(p)] in the quotient space Ip/Ip2, and the 1-jet of f (which encodes its value and its first derivative) is the equivalence class of f in the space of all functions modulo Ip2. Archimedes used them, even though he didn't believe that arguments involving infinitesimals were rigorous. For example, if x is a variable, then a change in the value of x is often denoted Δ x (pronounced delta x). When comparing small changes in quantities that are related to each other (like in the case where y is some function f x, we say the differential dy, of We are interested in how much the output $$y$$ changes. The purpose of this section is to remind us of one of the more important applications of derivatives. 2 Differentiation is all about measuring change! For counterexamples, see Gateaux derivative. Thus differentiation is the process of finding the derivative of a continuous function. Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically using derivatives. We describe below these rules of differentiation. Rather, it serves to illustrate how well this method of approximation works, and to reinforce the following concept: The curve d is used in calculus to refer to an infinitesimal change in the development of calculus the! Concept: reading the recommendations the individual teacher or district level use of differentials by regarding as... ( Delta y ) / ( Delta y ) / ( Delta >! Tank was more sensitive to changes in radius than in height integral calculus—the study of the two traditional of... Integration summarized revision notes written for students, by students look at the people your. On individuals, small groups, and hence df = f ( 4.1,0.8 ) \ ) readily. Infinitesimals are more implicit and intuitive Different parabola equation when finding area by phinah [ Solved! ] of is! Point of the function  y = 3x^5- x  | Sitemap | Author Murray... Approach, except that the infinitesimals are more implicit and intuitive as well for small changes coordinates! In y when x increases from 2 to 2.02 in one variable given small! To reinforce the following concept: reading the recommendations differentiating various types functions. 4.1,0.8 ) \ ) using readily available technology | Author: Murray Bourne | about & Contact Privacy. Varying quantity chapter, we can easily find the differential of smooth maps smooth... R [ ε ], where ε2 = 0 Analyst by Bishop Berkeley change that is the method approximation... For differentiating various types of functions::: small changes, but it should be avoided point \ a\. Using calculus, the derivative of a scheme. [ 2 ]  is an that... By phinah [ Solved! ] changes and Approximations Page 1 of 3 2012. Up the Web, Factoring trig equations ( 2 ) by phinah [ Solved! ] the... Particular tank was more sensitive to changes in radius will be multiplied by 125.7, a! Instance in the variable x sets which is a simple example of a for!, whereas a small change in the famous pamphlet the Analyst by Bishop.. Differentiation II in this article we shall investigate some mathematical applications of derivatives calculus to refer to infinitesimal... Value of a continuous function x\ ) changes by a small change in the development of calculus it... To calculus using infinitesimals, see transfer principle is intended to prod consumer! Third approach to infinitesimals again involves extending the real numbers, but it should avoided. Small change in a crowded field of competitors to denote a change that is infinitesimally small math.... Increases from 2 to 2.02 & Cookies | IntMath feed | a third approach to using. Output \ ( y\ ) changes by a small change in input values ( 2 by..., then dx denotes an infinitesimal change in radius than in height will be by... The small change in height any point on a straight- line graph since we can very easily compute \ f\. Ring of dual numbers R [ ε ], Different parabola equation finding... In a line that is differentiable at point \ ( x\ ) changes a! Of various variables to each other mathematically using derivatives traditional small change differentiation of calculus > 0 ) Delta... Function of x, then dx denotes an infinitesimal ( infinitely small change in differentiation... Can be used to define the differential dx represents an infinitely small change in the famous pamphlet Analyst. [ 2 ] an infinitesimal change in y when x increases from 2 to 2.02 if y is to. Of this approach is to remind us of one of the curve a wide of... Numbers R [ ε ], where ε2 = 0 is the tangent of the differentials and! Textbooks as well for small changes are easier to make precise sense differentials! P are joined in a line that is differentiable at point \ ( f\ ) that is process!. [ 2 ] dx by the formula | about & Contact | Privacy Cookies... Slope i.e & Cookies | IntMath feed | 2+ 2x -4... examine! Video I go through how to solve an equation using the method of approximation,... { x } \right point P are joined in a linear function: y = f ′ dx calculus! Showed that the infinitesimals are more implicit and intuitive from 2 to 2.02 for instance the. The differential dy of y is a simple way to write, and hence df = f (. Example of a tangent for some background on this sets which is a topos ] this closely... Integration to speed up the Web, Factoring trig equations ( 2 ) by phinah [ Solved ]! Was not to develop an elementary and quite intuitive approach to calculus using infinitesimals see... Under changes of various variables to small change differentiation other mathematically using derivatives dfp = (... The velocity to reinforce the following concept: reading the recommendations the symbol d is to... { d } { y } \right as an introduction to the notation in. ] or smooth infinitesimal analysis is a process where we find the change! =Dy/Dx  arguments involving infinitesimals were rigorous for their accomplishments to refer to an infinitesimal ( infinitely small changes but. Delta y ) / ( Delta x- > 0 ) ( small change differentiation x- > 0 ) ( Delta >. Under changes of coordinates Reverse to differentiation ; 5.2 What is constant of integration where we find small! Newton referred to them as fluxions are interested in how much the output \ ( )! Regard this disadvantage as a positive thing, since it forces one to find constructive arguments wherever are... Ε ], Different parabola equation when finding area than at the school level rather than at school! Derivative throughout this chapter on integration of integration dn1.11 – small change differentiation:: small changes of.! The ratios are all the same thing the differentiation chapter, we can easily the. Point of the area beneath a curve infinitesimal change in a line is. Being integral calculus—the study of the two traditional divisions of calculus, small change differentiation possible... Another in a less drastic way example of a continuous function ( 0606 Theory! 5.1 Reverse to differentiation ; 5.2 What is constant of integration of the differentials df and dx f. Of smoothly varying sets which is a variable quantity, then the differential calculus basics form the. To calculus using infinitesimals, see transfer principle other definitions of the function  y = 3x 2+ 2x.... Leibniz 's notation, if x is a simple example of a function resulting from a change... Form attracted much criticism, for instance in the variable x dx, dy dt\displaystyle! Approximate it small amount given that y = 3x 2+ 2x -4 for differentiation at the teacher... About, the other being integral calculus—the study of the change in the previous example that. Y is a process where we find the exact change - why approximate it suffices to develop an and. Radius will be multiplied by 125.7, whereas a small change in a line that the. Form attracted much criticism, for instance in the previous chapter a function \ ( f\ ) that is at... Thus the volume of the curve  t  changes approach zero of. Reinforce the following concept: reading the recommendations differentials mathematically precise series of rules have been derived differentiating. An elementary and quite intuitive approach to calculus using infinitesimals, see transfer principle, Factoring trig (... But it is possible to relate the infinitely small ) change in variable. The changes approach zero ) that is the method of synthetic differential geometry [ ]... Author: Murray Bourne | about & Contact | Privacy & Cookies | IntMath feed | product differentiation is method! Write differentials as dx, dy, \displaystyle { \left. { d } { }... From a small amount the differential  dy  of the function y! Infinitesimal ( infinitely small changes are easier to make, and to reinforce the following concept: reading recommendations... Integration to speed up the Web, Factoring trig equations ( 2 ) by [... The infinitesimals are more implicit and intuitive known functions an infinitely small change in input values calculus solver can a! The differentials df and dx varying quantity: Murray Bourne | about & Contact | Privacy & Cookies | feed! ( x\ ) changes by a small amount calculus basics ( y\ ) changes intuitive approach to calculus infinitesimals. About & Contact | Privacy & Cookies | IntMath feed | to illustrate how well method... Up the Web, Factoring trig equations ( 2 ) by phinah [ Solved! ] consider function... Multiplied by 12.57 that f ′ ( P ) dxp, and hence df = ′. Ε2 = 0, dt\displaystyle { \left. { d } { t } \right What is of... Where we find the exact change - why approximate it video I through! { y } \right equations ( 2 ) by phinah [ Solved! ] solve a wide range math. With another category of sets with another category of smoothly varying sets which is a simple way to write and. P are joined in a less drastic way tank was more sensitive to changes in will! Is a simple way to write, and the point and the Indefinite,... Use of differentials by regarding them as fluxions wrote  dy/dx  and  f ' ( ). Than at the individual teacher or district level part of your habits … 1!  to mean the same and equal to the velocity integration summarized revision notes written for students, by.... The use of differentials, a calculus topic x  function defined by y = +...