Mathematics
Differentiation from First Principles
Differentiation from first principles is a method used to find the derivative of a function by applying the definition of the derivative directly. It involves taking the limit of the difference quotient as the interval over which it is calculated approaches zero. This approach provides a fundamental understanding of how derivatives are calculated and is often used to derive key differentiation rules.
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6 Key excerpts on "Differentiation from First Principles"
eBook - PDF- Brian H. Chirgwin, Charles Plumpton(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
CHAPTER II THE TECHNIQUE OF DIFFERENTIATION 2:1 Differentiation from First Principles If y = j (χ) 9 we define the rate of increase or rate of change of y with f(x + h) -f(x) respect to x as lim provided this limit exists. This rate of change is variously denoted by /' (x) , Oy or —— , (d y/ά χ) , dx and is called the differential coefficient or derivative of y with respect to (w.r. to) x. The process of finding the derivative is called differentiation. We now find the derivatives of (i) x n , (ii) sin x . (i) f{x)=x»; ,. (x + h) n — x n Λ . (x + h) n — x n Λ , ft , fix) = lim v ^ = lim v , 7 = nx*' 1 (2.1) ' V ' A -> O A A+O (x + h)-x v ' on using eqn. (1.18) with x -- h and a? written in place of x and a respectively. (ii) / (x) = sin x ; „ , x ,. sin (a? + Ä) — sin x 2 sin I h cos (x + -I h) f (χ) = hm = lim ■ — A ->· o Ä -> o «-= lim —-y— · lim cos (x + -},-Ä) = cos x (2.2) on using eqn. (1.19) with Ih in place of x. Similarly d(cos x) dx = —sino;. (2.3) We now use a slightly different approach. Suppose y = f(x) and x increases to x + ôx. Here <5# stands for a small increment in x and § 2 : 1 THE TECHNIQUE OF DIFFERENTIATION 33 is not ô multiplied by x . Then y increases to y + δ y , where δ y is the increment in y corresponding to the increment òx in x, i.e., y + by = f(x + δχ).
eBook - PDFMathematical Methods
A Course of Mathematics for Engineers and Scientists
- Brian H. Chirgwin, Charles Plumpton(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
CHAPTER II THE TECHNIQUE OF DIFFERENTIATION 2 : 1 Differentiation from First Principles If y = /(x), we define the rate of increase or rate of change of y % . f(x + h)-f(x) . Λ Λ Λ with respect to x as hm - — provided this limit exists. dv This rate of change is variously denoted by /'(x), Oy or —-, ax (dy/dx), and is called the differential coefficient or derivative of y with respect to (w.r. to) x. The process of finding the derivative is called differentiation. We now find the derivatives of (i) x, (ii) sin x. (i)/(x) = x»; f( x ) = hm ^ L = h m > = nx n-i (2.1) A — o Λ Λ-^Ο (x+n)—x on using eqn. (1.19) with x+h and x written in place of x and a respectively. (ii) Assuming that term by term differentiation of an infinite power series is legitimate (see § 5:1), it follows at once from defi-nitions (1.1) and (1.2) for sin x, cos x that similarly, d (sin x) —~ = cos x, dx d (cos x) — ^ = —sin x. (2.2) (2.3) 69 CMES1-F
eBook - PDF- Doug French(Author)
- 2004(Publication Date)
- Continuum(Publisher)
Chapter 12 The Calculus: Differentiation and Integration There is no part of mathematics for which the methods of approach and development are more important than the calculus ... the early development must be gradual ... it will be found that pupils who have learnt to apply processes mechanically are mystified by the principles, and are therefore liable to serious error in any matter that is slightly outside the usual routine. (The Teaching of Calculus in Schools, Mathematical Association, 1951) ALGEBRA AND CALCULUS Since the calculus is far too big a subject to discuss in a single chapter in a book on algebra, I have chosen to use this final chapter to exemplify the ideas about teaching and learning algebra developed in this book by discussing some selected aspects of the calculus and its application to problems. Derivatives and integrals are concerned with two very important ideas: the gradient of a curve, with its links to rate of change, and the area under a curve with its links to summation. It is customary to introduce derivatives before integrals, but practice varies with regard to the approach to integration. One approach is to define it as an anti-derivative and then to show that it can be used to determine the area under a curve. The idea of an anti-derivative is obviously relevant when it comes to differential equations, but it does not seem to be the obvious starting point for considering integrals because it lacks an immediate purpose. The other approach is to define integrals directly in terms of the area under a curve and then let it emerge that the process of finding an integral is the reverse of finding a derivative. This second approach has the considerable merit of letting the link between differentiation and integration, the fundamental theorem of the calculus, emerge as a surprise.
eBook - PDFMathematics for Electronic Technology
Pergamon International Library of Science, Technology, Engineering and Social Studies
- D. P. Howson(Author)
- 2013(Publication Date)
- Pergamon(Publisher)
C H A P T E R I INTRODUCTORY CONCEPTS STUDENTS and engineers will come to a textbook such as this with widely differing mathematical training and skills. It therefore seemed useful to devote an introductory chapter to a range of topics, some of which may be familiar to the reader but all of which are indispensable if a proper grounding in the subject is to be ob-tained. As in the rest of the book, a number of examples of varying degrees of difficulty are included and solutions, at least in outline, are usually provided. The reader should make a point of working through these to improve the grasp of the subject, particularly as the text treatment of each topic is necessarily brief in a little book like this. Nevertheless, it has been thought worth while to com-mence with a formal grounding in differentiation and integration, partly to refresh the mind as to the precise meanings of these con-cepts, but also to allow comparison of the definitions with those used as the subject is developed in other sections of the work. The last part of the chapter is devoted to vector theory, commencing with elementary material, but proceeding to a relatively advanced level. 1.1 Differentiation The idea of the rate of change of a variable is basic to a study of many engineering problems, and it will be assumed that the student is familiar with this. Here only the salient points of the theory will be covered, and a summary provided of some of the most important results. The differential, or rate of change, of a function of x, f(x), with 1 2 M A T H E M A T I C S FOR E L E C T R O N I C T E C H N O L O G Y respect to an infinitesimal change in x, will be denoted by fx) or d//dx and defined at a point x 0 by idf = l jm f/(*o + Sx) - f(x 0 )} (1) f(x) will accordingly be said to be differentiable at x 0 if such a limit exists. This will occur if /(x) is continuous at x 0 , in other words if lim/(x)=/(x 0 ) X-+XO (2) independently of the way in which χ approaches x 0 .
eBook - PDFFree Calculus: A Liberation From Concepts And Proofs
A Liberation from Concepts and Proofs
- Qun Lin(Author)
- 2008(Publication Date)
- World Scientific(Publisher)
Chapter 1 Official Calculus: Differentiation and Integration Outline: Chapter 1 = two inequalities = differential inequality + fundamental inequality. The latter is the sum of the former (over all subintervals). And hence, finally, Chap. 1 = one inequality. Calculus in this book only admits a definition of the differential [or derivative, see (1.1)] without other prior knowledge. The FT, in one word, is the sum of these definitions without more proofs. This has been described in using plain English in the first part of the Preface. This chapter will use the function language in order to get more conclusions (e.g., the TT) and applications. 1.0. A Case: Height and Slopes Trigonometry begins with measuring the height (unknown) of a tree using a slope (what we know) or the direction of a dummy hypotenuse. This problem leads to a tangent formula. A regular triangle. 15 16 Free Calculus—A Liberation from Concepts and Proofs Parallely, calculus begins with measuring the height of a hill using the slopes of the curve hillside, as a curved trig (see Refs. 15–20). This leads to the Newton–Leibniz formula. A curved triangle. But calculus need more concepts e.g. the tangent slope, differential, measured error, relative error and total error, as indicated in the Preface with the following figures: we translate them into the function languages as follows. Official Calculus: Differentiation and Integration 17 1.1. Translating into Function Language Let f = f ( x ) be a given function (producing a curve) defined on the interval [ a , b ] . Then, over each subinterval [ x , x + h ] , for all points near the node x . True height: f ( x + h ) − f ( x ) Tangent slope tan θ 0 at node x : Derivative f ( x ) Differential: f ( x ) h Each measured error = (relative error)(base): f ( x + h ) − f ( x ) − f ( x ) h = ( x , h ) h If over each subinterv [ x , x + h ] ( x , h ) 1 (1.1) the derivative f is uniquely define pointwisely.
eBook - PDF- Tom M. Apostol(Author)
- 2019(Publication Date)
- Wiley(Publisher)
5 THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION 5.1 The derivative of an indefinite integral. The first fundamental theorem of calculus We come now to the remarkable connection that exists between integration and differentiation. The relationship between these two processes is somewhat analogous to that which holds between “squaring” and “taking the square root.” If we square a positive number and then take the positive square root of the result, we get the original number back again. Similarly, if we operate on a con- tinuous function f by integration, we get a new function (an indefinite integral of f ) which, when differentiated, leads back to the original function f . For example, if f (x) = x 2 , then an indefinite integral A of f may be defined by the equation A(x) = ∫ x c f (t) dt = ∫ x c t 2 dt = x 3 3 − c 3 3 , where c is a constant. Differentiating, we find A ′ (x) = x 2 = f (x). This example illustrates a general result, called the first fundamental theorem of calculus, which may be stated as follows: theorem 5.1. first fundamental theorem of calculus. Let f be a function that is integrable on [a, x] for each x in [a, b]. Let c be such that a ≤ c ≤ b and define a new function A as follows: A(x) = ∫ x c f (t) dt if a ≤ x ≤ b. Then the derivative A ′ (x) exists at each point x in the open interval (a, b) where f is continuous, and for such x we have A ′ (x) = f (x). (5.1) First we give a geometric argument which suggests why the theorem ought to be true; then we give an analytic proof. 202 The derivative of an indefinite integral. The first fundamental theorem of calculus 203 Geometric motivation. Figure 5.1 shows the graph of a function f over an interval [a, b]. In the figure, h is positive and ∫ x+h x f (t) dt = ∫ x+h c f (t) dt − ∫ x c f (t) dt = A(x + h) − A(x). The example shown is continuous throughout the interval [x, x + h]. Therefore, by the mean-value theorem for integrals, we have A(x + h) − A(x) = hf (z), where x ≤ z ≤ x + h.
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