Combining Differentiation Rules

4 Key excerpts on "Combining Differentiation Rules"

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  eBook - ePub

  Mathematical Methods for Finance

  Tools for Asset and Risk Management

  • Sergio M. Focardi, Frank J. Fabozzi, Turan G. Bali(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  x = 0 is continuous but not differentiable. However, there are examples of functions that defy visual intuition; in fact, it is possible to demonstrate that there are functions that are continuous in a given interval but never differentiable. One such example is the path of a Brownian motion which we will discuss in Chapter 10.

  FIGURE 2.3 Geometric Interpretation of a Derivative

  COMMONLY USED RULES FOR COMPUTING DERIVATIVES

  There are rules for computing derivatives. These rules are mechanical rules that apply provided that all derivatives exist. The proofs are provided in all standard calculus books. The basic rules are:

  Rule 1 : (c ) = 0, where c is a real constant.

  Rule 2 : (bx n ) = nbx

  n − 1

  , where b is a real constant.

  Rule 3 : (af (x ) + bg (x )) = a f (x ) + b g (x ), where a and b are real constants.

  Rule 3 is called the rule of termwise differentiation and shows that differentiation is a linear operation. Let’s apply the basic rules to the following function:

  where a , b 1 , b 2 , b 3 , …,

  bn

  are the constants.

  The first term is just a and as per Rule 1 the derivative is zero. The derivative of b 1 x by Rule 2 is b 1 . For each term

  bi xi

  by Rule 2 the derivative is

  ibi xi

  −1 . Thus, the derivative of

  Therefore, the derivative of y is

  There is a special rule for a composite function . Consider a composite function: h (x ) = f [g (x )]. Provided that h and g are differentiable at the point x and that f is derivable at the point s = g (x ), then the following rule, called the chain rule , applies:

  Let’s take the first derivative of the following composite function: where and Applying the chain rule gives

  Table 2.2 shows the sum rule, product rule , quotient rule , and chain rule for calculating derivatives in both standard and infinitesimal notation. In Table 2.2 , it is assumed that a , b are real constants (i.e., fixed real numbers), that f , g , and h are functions defined in the same domain, and that all functions are differentiable at the point x . Table 2.3

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  eBook - PDF

  Finite Mathematics and Applied Calculus

  • Stefan Waner, Steven Costenoble(Authors)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  TECHNIQUES OF DIFFERENTIATION WITH APPLICATIONS Projecting Market Growth It is 2010, and you are on the board of directors at Fullcourt Academic Press . The sales director of the high school division has just burst into your office with a proposal for an expansion strategy based on the assumption that the number of graduates from private high schools in the United States will grow at a rate of at least 4,000 per year through the year 2015. Because the figures actually appear to be leveling off, you are suspicious about this estimate. You would like to devise a model that predicts this trend before tomorrow’s scheduled board meeting. How do you go about doing this? Yuri Arcurs/Shutterstock.com 821 11.1 Derivatives of Powers, Sums, and Constant Multiples 11.2 A First Application: Marginal Analysis 11.3 The Product and Quotient Rules 11.4 The Chain Rule 11.5 Derivatives of Logarithmic and Exponential Functions 11.6 Implicit Differentiation KEY CONCEPTS REVIEW EXERCISES CASE STUDY www.WanerMath.com At the Website, in addition to the resources listed in the Preface, you will find: The following extra topic: • Linear Approximation and Error Estimation CASE STUDY Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 822 Chapter 11 Techniques of Differentiation with Applications Introduction In Chapter 10 we studied the concept of the derivative of a function, and we saw some of the applications for which derivatives are useful. However, computing the deriva-tive of a function algebraically, from the definition, seemed to be a time-consuming process, forcing us to restrict attention to fairly simple functions. In this chapter we develop shortcut techniques that will allow us to write down the derivative of a function directly without having to calculate any limit.

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  eBook - PDF

  Applied Calculus

  • Stefan Waner, Steven Costenoble(Authors)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  TECHNIQUES OF DIFFERENTIATION WITH APPLICATIONS Projecting Market Growth It is 2010, and you are on the board of directors at Fullcourt Academic Press. The sales director of the high school division has just burst into your office with a proposal for an expansion strategy based on the assumption that the number of graduates from private high schools in the United States will grow at a rate of at least 4,000 per year through the year 2015. Because the figures actually appear to be leveling off, you are suspicious about this estimate. You would like to devise a model that predicts this trend before tomorrow’s scheduled board meeting. How do you go about doing this? Yuri Arcurs/Shutterstock.com 303 4.1 Derivatives of Powers, Sums, and Constant Multiples 4.2 A First Application: Marginal Analysis 4.3 The Product and Quotient Rules 4.4 The Chain Rule 4.5 Derivatives of Logarithmic and Exponential Functions 4.6 Implicit Differentiation KEY CONCEPTS REVIEW EXERCISES CASE STUDY www.WanerMath.com At the Website, in addition to the resources listed in the Preface, you will find: The following extra topic: Linear Approximation and Error Estimation CASE STUDY Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 304 Chapter 4 Techniques of Differentiation with Applications Introduction In Chapter 3 we studied the concept of the derivative of a function, and we saw some of the applications for which derivatives are useful. However, computing the deriva- tive of a function algebraically, from the definition, seemed to be a time-consuming process, forcing us to restrict attention to fairly simple functions. In this chapter we develop shortcut techniques that will allow us to write down the derivative of a function directly without having to calculate any limit.

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  eBook - PDF

  Applied Calculus

  for Business, Life, and Social Sciences

  • Denny Burzynski(Author)
  • 2014(Publication Date)
  • XYZ Textbooks

    (Publisher)

  © Dean Bertoncelj/iStockPhoto 2 131 Chapter Outline 2.1 The Derivative of a Function and Two Interpretations 2.2 Differentiating Products and Quotients 2.3 Higher-Order Derivatives 2.4 The Chain Rule and General Power Rule 2.5 Implicit Differentiation P eople tend to drink more water on hot days than they do on cold days. Because most drinking water is sold in plastic bottles that can be recycled, as the tem- perature goes up, there is an increase in the number of plastic bottles recycled. We can model this situation with the following diagram. Once we have the function for these quantities, we can differentiate them, giving us some rates of change. Further, we have two types of questions we can answer, one using just algebra, and the other using calculus. Algebra How many plastic bottles are recycled when the temperature is 85 °F? Calculus How fast is the number of plastic bottles recycled changing when the temperature is 85 °F? As you might expect, this situation involves a composite function because the number of plastic bottles recycled depends on the number of plastic bottles sold, which in turn, depends on the temperature. Among the rules we will develop in this chapter is the rule for differentiating a composite function. Depends on Depends on The number of water bottles sold Temperature The number of plastic bottles recycled R S T Differentiation: The Language of Change Note When you see this icon next to an example or problem in this chapter, you will know that we are using the topics in this chapter to model situations in the world around us. 132 Study Skills If you have successfully completed Chapter 1, then you have made a good start at developing the study skills necessary to succeed in all math classes. Some of the study skills for this chapter are a continuation of the skills from Chapter 1, while others are new to this chapter. 1. Continue to Set and Keep a Schedule Sometimes I find students do well in Chapter 1 and then become overconfident.

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