Differentiation Rules

7 Key excerpts on "Differentiation Rules"

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  Mathematics for Business Analysis

  • Paul Turner, Justine Wood(Authors)
  • 2023(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  rules for differentiation , which can be applied across a wide range of functions of interest.

  The rules of differentiation provide a set of results that allow us to find the derivatives of many functions without needing to use first principles. Since the method of first principles is not always easy to apply, these rules can save us a great deal of time and effort. Therefore, we will therefore set out some of the more important rules below, along with proofs and examples where it is useful.

  Rule 1: Multiplication by a Constant

  Consider a function defined by the equation and let , where a is a real number. The derivative of y with respect to x is equal to the derivative of u with respect to x multiplied by the same constant, that is

  Proof: This rule is easily proved using the increment theorem. Since , we have , where is infinitesimal. The derivative is therefore

  .

  EXAMPLE

  We have already shown that, for , . Therefore, if we define a new function of the form it follows that

  Rule 2: Sum–Difference Rule

  Let and . If we now define a new function as the sum, or difference, of these functions, that is either or , then the derivative of this new function will be either the sum or the difference of the derivatives of the original function. That is, if then and if then The proof of this rule is obvious and is left as an exercise for the interested reader.

  EXAMPLE

  If then by the sum–difference rule

  Rule 3: The Product Rule

  Let and . Let then the derivative of y with respect to x is given by the following expression

  .

  The proof of this rule is a little trickier than that for the sum-difference rule and is set out explicitly below.

  Proof: Let be an infinitesimal change in the x variable. We have

  Since and have nonzero standard parts but the standard part of is equal to zero, taking the standard part of this expression yields

  which establishes the desired result. This is referred to as the product rule of differentiation

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  Foundations of Mathematics

  Algebra, Geometry, Trigonometry and Calculus

  • Philip Brown(Author)
  • 2016(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  8

  DIFFERENTIAL CALCULUS

  8.1INTRODUCTION

   

  Differential calculus, the topic of this chapter, is, in large part, the legacy of Sir Isaac Newton. In his gigantic publication, The Principia Mathematica, Newton introduced the mathematics of calculus and applied it to the scientific study of the orbits of the planets around the sun. Thus, he not only revolutionized science but also created the mathematics he needed in order to do so.

  This chapter deals with the derivatives of functions. While we study a function abstractly as mathematical object, in real-world applications it is a representation of motion (in a very general sense). In differential calculus, the notion of the derivative of a function, and the corresponding geometric notion of the slope of a tangent line to the graph of the function, relates to the idea of “instantaneous motion.” As we will begin to explain, in this chapter, the knowledge gained about the “instantaneous motion” of a function enables us to investigate and discover important properties of the function; for example, where the peaks of a function occur, or the approximate behavior of a function near any particular point in the domain of the function.

  In this chapter we will give the definition of the derivative, introduce the notion of a derivative function and present the rules (the power rule, sum rule, product rule, quotient rule and the chain rule) for computing it, and apply the definition and rules to compute the derivatives of the standard functions (such as polynomial, rational, root, trigonometric, exponential and logarithmic, inverse trig), and vector functions.

  In section 8.2 , the definition of the derivative is introduced in four parts: in section 8.2.1 , the notion of a graph “leveling out” at the origin and its mathematical statement in terms of a limit is an intuitive and mathematically simplest starting point. In section 8.2.2 , the tangent line to a graph at the origin is introduced as the best approximating line by means of a graphical example. This is interpreted in section 8.2.3 , as it is stated that the difference function (graph minus tangent line) “levels out” at the origin. Consequently, the limit formula from section 8.2.1 can be applied to derive the slope of the tangent line at the origin. Finally, in section 8.2.4

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  Mathematics N5 Student's Book

  TVET FIRST

  • JV John(Author)
  • 2022(Publication Date)
  • Macmillan

    (Publisher)

  2 Module 17 Differentiation TVET FIRST Differentiation Overview of Module 2 Differentiation is the process of finding the rate of change of a function at an instant and has numerous applications in engineering. Along with integral calculus, differential calculus is one of the two major branches of calculus. It is essential to have a good grasp of differentiation techniques before working with its applications. In this module, we will learn to differentiate using various methods, to work with and differentiate inverse trigonometric functions and to sketch their graphs. Differentiation Implicit and explicit functions Differentiating implicit functions 2.3: Implicit differentiation The concept of differentiation Differentiation from first principles: o Functions [ f (x) = ax n , n ≠ 1, 2, 3] using the binomial theorem o Fractions [ f (x) = ax + b _ cx + d ] o Trigonometric functions [ f (x) = sin x and f (x) = cos x] 2.1: Differentiation from first principles Deriving the derivatives of: tan x, cot x, sec x, cosec x Rules of differentiation (standard derivatives) Power rule, product rule, quotient rule Chain rule Combining the rules 2.2: Differentiation techniques 2.6: Mixed differentiation examples Summary of logarithmic laws Applying the method 2.4: Logarithmic differentiation Differentiation Sketching the graphs 2.5: Inverse trigonometric functions Note The full learning outcomes for each module are listed in the table at the back of the book. Figure 2.1: Differentiation is a powerful technique used by engineers Starter activity Read through section 2.1.1 as revision and try the following activity in groups: 1. Given f (x) = x + 2, find: 1.1 f (2) 1.2 f (–2) 1.3 f (x + 2) 1.4 f (h) 1.5 f (x + h) 2. Given f (x) = −x 2 + x − 2, find: 2.1 f (0) 2.2 f (–1) 2.3 f (x + h) 2.4 f (x + h) – f (x) 18 Module 2 TVET FIRST Unit 2.1: Differentiation from first principles You were introduced to the concept of differentiation and its applications in N4.

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  A Mathematics Course for Political and Social Research

  • Will H. Moore, David A. Siegel(Authors)
  • 2013(Publication Date)
  • Princeton University Press

    (Publisher)

  We’ll need to calculate derivatives for more complex functions in political science, however. Rather than go back to the definition each time, it will help to have some rules for the differentiation of specific functions and types of functions we can call on. This chapter presents those rules. A common method of presentation of this material provides a list of many seemingly disconnected rules. While that type of presentation provides clear rules and an easy memorization device, we feel that it can lead to the perception that calculus is somehow more complicated than it is, fraught with specificities that one must memorize. In reality, most of the commonly used rules stem from a handful of properties of the derivative operator, and most of these properties may be deduced from the definition. To help make these connections, we present the material accordingly. In Section 1 we develop the rules typically called the sum rule, product rule, quotient rule, and chain rule largely from the definition of the derivative itself. In Section 2 we use these rules and the definition of the derivative to offer deriva- tives of many common and special functions used in political science. In Section 3 we relent and provide a summary accounting of the rules for differentiation for ready reference, along with a discussion as to when to use each of the rules. If you already know or don’t care about calculus but need to calculate some derivatives, you can skip to that section. You can also skip to that section if you are finding the derivations of the rules too difficult to follow the first time through. We expect that many students who have not had calculus before will be in this boat, and might benefit from some experience manipulating deriva- tives before going back to see whence came the rules they used to perform the manipulations.

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  eBook - PDF

  Calculus: Early Transcendentals, Metric Edition

  • James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 188 CHAPTER 3 Differentiation Rules The Quotient Rule and the other differentiation formulas enable us to compute the derivative of any rational function, as the next example illustrates. EXAMPLE 4 Let y - x 2 1 x 2 2 x 3 1 6 . Then y9 - s x 3 1 6d d dx s x 2 1 x 2 2d 2 s x 2 1 x 2 2d d dx s x 3 1 6d s x 3 1 6d 2 - s x 3 1 6ds2x 1 1d 2 s x 2 1 x 2 2ds3x 2 d s x 3 1 6d 2 - s2x 4 1 x 3 1 12x 1 6d 2 s3x 4 1 3x 3 2 6x 2 d s x 3 1 6d 2 - 2x 4 2 2x 3 1 6x 2 1 12 x 1 6 s x 3 1 6d 2 ■ EXAMPLE 5 Find an equation of the tangent line to the curve y - e x ys1 1 x 2 d at the point (1, 1 2 e). SOLUTION According to the Quotient Rule, we have dy dx - s1 1 x 2 d d dx se x d 2 e x d dx s1 1 x 2 d s1 1 x 2 d 2 - s1 1 x 2 de x 2 e x s2xd s1 1 x 2 d 2 - e x s1 2 2x 1 x 2 d s1 1 x 2 d 2 - e x s1 2 xd 2 s1 1 x 2 d 2 So the slope of the tangent line at (1, 1 2 e) is dy dx Z x-1 - 0 This means that the tangent line at (1, 1 2 e) is horizontal and its equation is y - 1 2 e. (See Figure 4.) ■ NOTE Don’t use the Quotient Rule every time you see a quotient. Sometimes it’s easier to rewrite a quotient first to put it in a form that is simpler for the purpose of differenti- ation. For instance, although it is possible to differentiate the function Fs xd - 3x 2 1 2 sx x using the Quotient Rule, it is much easier to perform the division first and write the func- tion as Fs xd - 3x 1 2x 21y2 before differentiating. Figure 3 shows the graphs of the function of Example 4 and its deriva- tive.

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  Calculus of a Single Variable: Early Transcendental Functions, International Metric Edition

  • Ron Larson, Bruce Edwards(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  150 Chapter 3 Differentiation 3.3 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises. CONCEPT CHECK 1. Product Rule Describe the Product Rule in your own words. 2. Quotient Rule Describe the Quotient Rule in your own words. 3. Trigonometric Functions What are the derivatives of tan x, cot x, sec x, and csc x? 4. Higher-Order Derivative What is a higher-order derivative? Using the Product Rule In Exercises 5–10, use the Product Rule to find the derivative of the function. 5. g(x) = (2x - 3)(1 - 5x) 6. y = (3x - 4)(x 3 + 5) 7. h(t) = radical.alt2t (1 - t 2 ) 8. g(s) = radical.alt2s (s 2 + 8) 9. f (x) = e x cos x 10. g(x) = radical.alt2x sin x Using the Quotient Rule In Exercises 11–16, use the Quotient Rule to find the derivative of the function. 11. f (x) = x x - 5 12. g(t) = 3t 2 - 1 2t + 5 13. h(x) = radical.alt2x x 3 + 1 14. f (x) = x 2 2radical.alt2x + 1 15. g(x) = sin x e x 16. f (t) = cos t t 3 Finding and Evaluating a Derivative In Exercises 17–24, find f uni2032 parenleft.alt1xparenright.alt1 and f uni2032 parenleft.alt1cparenright.alt1. 17. f (x) = (x 3 + 4x)(3x 2 + 2x - 5), c = 0 18. f (x) = (2x 2 - 3x)(9x + 4), c = -1 19. f (x) = x 2 - 4 x - 3 , c = 1 20. f (x) = x - 4 x + 4 , c = 3 21. f (x) = x cos x, c = π 4 22. f (x) = sin x x , c = π 6 23. f (x) = e x sin x, c = 0 24. f (x) = cos x e x , c = 0 Using the Constant Multiple Rule In Exercises 25–30, complete the table to find the derivative of the function without using the Quotient Rule. Function Rewrite Differentiate Simplify 25. y = x 3 + 6x 3 26. y = 5x 2 - 3 4 Function Rewrite Differentiate Simplify 27. y = 6 7x 2 28. y = 10 3x 3 29. y = 4x 3H208622 x 30. y = 2x x 1H208623 Finding a Derivative In Exercises 31–42, find the derivative of the algebraic function. 31. f (x) = 4 - 3x - x 2 x 2 - 1 32. f (x) = x 2 + 5x + 6 x 2 - 4 33. f (x) = x parenleft.alt4 1 - 4 x + 3 parenright.alt4 34. f (x) = x 4 parenleft.alt4 1 - 2 x + 1 parenright.alt4 35.

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  Mathematics of Economics and Business

  • Frank Werner, Yuri N. Sotskov(Authors)
  • 2006(Publication Date)
  • Routledge

    (Publisher)

  4 Differentiation

  In economics, there are many problems which require us to take into account how a function value changes with respect to small changes of the independent variable (e.g. input, time, etc.). For example, assume that the price of some product changes slightly. The question is how does this affect the amount of product customers will buy? A useful tool for such investigations is differential calculus, which we treat in this chapter. It is an important field of mathematics with many applications, e.g. graphing functions, determination of extreme points of functions with or without additional constraints. Differential calculus allows us to investigate specific properties of functions such as monotonicity or convexity. For instance in economics, cost, revenue, profit, demand, production or utility functions have to be investigated with respect to their properties. In this chapter, we consider functions f : Df → ℝ depending on one real variable, i.e. Df ⊆ ℝ.

  4.1 LIMIT AND CONTINUITY

  4.1.1 Limit of a function

  One of the basic concepts in mathematics is that of a limit (see Definition 2.6 for a sequence). In this section, the limit of a function is introduced. This notion deals with the question of which value does the dependent variable y of a function f with y = f (x) approach as the independent variable x approaches some specific value x0 ?

  Definition 4.1   The real number L is called the limit of function f : Df → ℝ as x tends to x0 if for any sequence {xn } with xn ≠ x0 , xn ∈ Df , n = 1, 2, . . . , which converges to x0 , the sequence of the function values {f (xn )} converges to L.

  Thus, we say that function f tends to number L as x tends to (but is not equal to) x0 . As an abbreviation we write

  Note that limit L must be a (finite) number, otherwise we say that the limit of function f as x tends to x0 does not exist. If this limit does not exist, we distinguish two cases. If L = ±∞, we also say that function f is definitely divergent as x tends to x0 , otherwise function f is said to be indefinitely divergent as x tends to x0

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