Differentiation Rules
Differentiation rules are a set of guidelines used to find the derivative of a function. These rules include the power rule, product rule, quotient rule, and chain rule, among others. They provide a systematic approach to calculating derivatives and are fundamental in calculus for analyzing the rate of change of functions.
7 Key excerpts on "Differentiation Rules"
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...CHAPTER 4 Rules of Differentiation CHAPTER 4 RULES OF DIFFERENTIATION 4.1 INTRODUCTION It would be an extremely tedious process to always determine derivatives by the use of limits. As a result, derivative rules for a wide variety of functions can be established by using limits, and those rules become the more efficient tools for finding derivatives. Since this text is not intended to be a first exposure to calculus, most of the derivative rules will be given without proof, although a few key derivatives will be established through limits. 4.2 DERIVATIVES OF POLYNOMIALS Some derivatives can be determined just from the knowledge that a derivative represents the slope of the function. For instance, the derivative of a constant is zero, since the slope of the graph of y = c, where c is a constant, is 0. Derivative of a Constant If c is any constant, and y = c, then Using similar reasoning, the derivative of any linear function is the slope of that function. Derivative of a Linear Function If m is any constant, and y = mx, then The derivative of a power of x is also relatively straightforward, but perhaps not intuitively understood. If m is any natural number, and f (x) = x m then The proof of this rule utilizes the definition of the derivative and the binomial expansion pattern from a prerequisite course. Although the binomial expansion from courses prior to calculus applies only to exponents that are natural numbers, we will extend the derivative rule (without proof) to any exponents that are elements of the real numbers. Derivative of a Power If m is any real number, and f (x) = x m, then The derivative of any constant multiple of a power of x is established by factoring the constant out of the limit. Therefore, the coefficient simply multiplies by the power. Derivative of a Constant Times a Function If c and m are constants and y = cx m, then Extending derivatives of monomials to polynomials is also elementary...
eBook - ePub- Arsen Melkumian(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...4 Limits and derivatives In mathematics, differential calculus is a subfield of calculus that is concerned with the study of how quickly functions change over time. The primary concept in differential calculus is the derivative function. The derivative allows us to find the rate of change of economic variables over time. This chapter introduces the concept of a derivative and lays out the most important rules of differentiation. To properly introduce derivatives, one needs to consider the idea of a limit. We cover the concept of a limit in the first section. The chapter closes with growth rates of discrete and continuous variables. 4.1 Limits Consider a function g given by and shown in Figure 4.1. Clearly, the function is undefined for x = 0, since anything divided by zero is undefined. However, we can still ask what happens to g (x) when x is slightly above or below zero. Using a calculator we can find the values of g (x) in the neighborhood of x = 0, as shown in Table 4.1. As x approaches zero, g (x) takes values closer and closer to 2. So we can say that g(x) tends to 2 as x tends to zero. We write and say that the limit of g (x) as x approaches zero is equal to 2. Now that the idea of a limit is clear on an intuitive level, let us consider a formal definition of the right- and left-hand side limits. Let f be a function defined on some open interval (a, b). We say that L is the right-hand side limit of f (x) as x approaches a from the right and write if for every ε > 0 there is a δ > 0 such that Figure 4.1 Table 4.1 whenever As an example, let us consider the following function We want to show that Let us choose ε > 0. We need to show that there is a δ > 0 such that whenever Let us choose δ = (ε/ 2). Then, and therefore It follows immediately that whenever Now we have proved that the limit of as x approaches zero from the right is equal to 1. Now let us define a left-hand side limit. Let f be a function defined on some open interval (a, b)...
eBook - ePub- Adil H. Mouhammed(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...Chapter Two Derivatives and Applications Economic decisions are based on marginal analysis. For example, the monopolist’s best level of output is determined by equating marginal cost and marginal revenue. To find the marginal cost and revenue, total cost and revenue functions must be differentiated with respect to the output level. Similarly, derivatives can be used in many applications in business and economics. For this reason, the rules of differentiation are outlined in this chapter, and many applications are provided. The Concept of Derivative The derivative of a function measures the rate of change of the dependent variable y with respect to the independent variable x--the slope of the function. That is, the derivative indicates the impact of a small change in x on y. For example, suppose the dependent variable y is the quantity supplied by a producer, and x is the price of that product. Mathematically, the function is written as y = f (x). Now, if the price x changes by a very small amount (dx), the quantity supplied will change by a very small amount (dy) as well. These small changes, dx and dy, are called the differential of x and y, respectively. After these changes, the new magnitude of the two variables becomes (y + dy) and (x + dx). And dy/dx is called the derivative of y with respect to x. In other words, dy/dx shows the changes in y per unit change in x. The process of finding the derivative is called the differentiation process. If a given function is a univariate function, such as the above, the following rules of differentiation (Glaister 1984; Chiang 1984; Ostrosky and Koch 1986) are applied: Rule 1: Derivative of a Constant Function If y = f(x) = k, where k is a constant, then dy/dx = 0 Example 1: Differentiate y = f(x)= 30. Solution: dy/dx = 0 Example 2: If the fixed cost of a product q is FC = 20, then the derivative of the fixed cost with respect to q is d(FC)/dq = 0...
eBook - ePub- Adil H. Mouhammed(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...CHAPTER TWO Derivatives and Applications Economic decisions are based on marginal analysis. For example, the monopolist's best level of output is determined by equating marginal cost and marginal revenue. To find the marginal cost and revenue, total cost and revenue functions must be differentiated with respect to the output level. Similarly, derivatives can be used in many applications in business and economics. For this reason, the rules of differentiation are outlined in this chapter, and many applications are provided. The Concept of Derivative The derivative of a function measures the rate of change of the dependent variable y with respect to the independent variable x--the slope of the function. That is, the derivative indicates the impact of a small change in x on y. For example, suppose the dependent variable y is the quantity supplied by a producer, and x is the price of that product. Mathematically, the function is written as y = f(x). Now, if the price x changes by a very small amount (dx), the quantity supplied will change by a very small amount (dy) as well. These small changes, dx and dy, are called the differential of x and y, respectively. After these changes, the new magnitude of the two variables becomes (y + dy) and (x + dx). And dy/dx is called the derivative of y with respect to x. In other words, dy/dx shows the changes in y per unit change in x. The process of finding the derivative is called the differentiation process. If a given function is a univariate function, such as the above, the following rules of differentiation (Glaister 1984; Chiang 1984; Ostrosky and Koch 1986) are applied: Rule 1: Derivative of a Constant function If y = f(x) = k, where k is a constant, then dy/dx = 0 Example 1: Differentiate y = f(x) = 30. Solution: dy/dx = 0 Example 2: If the fixed cost of a product q is FC = 20, then the derivative of the fixed cost with respect to q is d(FC)/dq = 0...
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...CHAPTER 7 Calculus: Expanding the Toolkit Figure 7.1 A 3-day-old chick embryo. Image courtesy of Nobue Itasaki, Division of Developmental Neurobiology, MRC National Institute for Medical Research. Figure 7.2 In the wild, crocodiles regulate their body temperature by basking in water when they need to cool off, and then shuttling back to the land to warm up in the sunshine. Image courtesy of Mister-E under Creative Commons Attribution 2.0 Generic. Chapters 5 and 6 introduced the basic equipment you need to construct new theories by analyzing the behavior of curves. To push the analogy with a builder’s toolkit rather too far, we could view differentiation as supplying a metaphorical ‘spirit level’ that is sensitive to the gradient of a function, whereas integration provides a trowel for spreading mortar over a range of different areas. The aim of this chapter is to describe a set of ‘power tools’ that will make it possible for you to perform tasks that couldn’t be accomplished ‘by hand’ using the methods encountered up to this point. For differentiation, the chain rule, the product rule, and the quotient rule are vital for slicing your way through an array of more complex functions. Similarly, the rule for changing the variable of an integral provides just the sort of heavy lifting gear that is necessary for shifting weighty integration problems. These new techniques are put to work to investigate a collection of biological topics, from the metabolism of reptiles to the size of rabbits, by way of weak acids, population dynamics, and the forces experienced by the foot of a sprinting athlete. Periodic functions appear regularly as applications, culminating in a discussion of simple harmonic motion at the end of the chapter. 7.1 Sinusoidal functions The trigonometric functions discussed in Chapter 4 are useful for modeling biological data that vary in a repetitive way as a function of time...
- J. Rosebush, Stu Schwartz(Authors)
- 2016(Publication Date)
- Research & Education Association(Publisher)
...So, if equals one of these indeterminate forms, then take. Note that you are not using the quotient rule here, you are simply taking the derivative of the numerator and denominator separately. If the limit still has an indeterminate form then repeat the process as necessary. This also applies to cases in which x → ±∞. 1. For example,. Using L’Hôspital’s rule,. G. Derivative Rules 1. When taking the derivative of a function you might have to use more than one of the above rules. 2. There are some functions whose derivatives occur very often on the exam and it would save you time if memorized. These are the derivatives of and more generally, ; and and more generally,. Note that the chain rule was used in both general cases. H. Derivatives of trigonometric functions 1. The derivatives of the cofunctions are negative. 2. In taking the derivative of most trigonometric functions you will need to use the chain rule since most will be compositions—sometimes of more than two functions. Here is an example of the derivative of a function of the form y = f (g (h)): y = sin(tan(x 2)) → y ′ = cos(tan(x 2))sec 2 (x 2)(2 x). I. Derivatives of inverse trigonometric functions 1. Note that the derivatives of the cofunctions are the negatives of the derivatives of the functions. 2. In most cases, the chain rule is used. For example,. J. Implicit Differentiation—this means finding y ′ when the equation given is not explicitly defined in terms of y (that is, it is not of the form y = f (x)). In this case you must remember to always use the chain rule when taking the derivative of an expression involving y. That is all! Example 1 : Find y ′ if x 2 + y 2 = 3. Taking derivatives on both sides, 2 x + 2 yy ′ = 0 →. Example 2: Find y ′ if x 2 y 2 – 3 ln y = x + 7. Taking derivatives on both sides,...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...10 Differentials, Derivatives, and Partial Derivatives The concept of differential entails a small (tendentially negligible) variation in a variable x, denoted as dx, or a function f { x }, denoted as df { x }; the associated derivative of f { x } with regard to x is nothing but the ratio of said differentials, i.e. df / dx – usually known as Leibnitz’s formulation. In the case of a bivariate function, say, f { x,y }, differentials can be defined for both independent variables, i.e. dx and dy – so partial derivatives will similarly arise, i.e. ∂f / ∂x and ∂f / ∂y ; operator ∂ is equivalent to operator d, except that its use is exclusive to multivariate functions – in that it stresses existence of more than one independent variable. 10.1 Differential In calculus, the differential represents the principal part of the change of a function y =. f { x } – and its definition reads (10.1) where df / dx denotes the derivative of f { x } with regard to x ; it is normally finite, rather than infinitesimal or infinite – yet the precise meaning of variables dx and df depends on the context of application, and the required level of mathematical accuracy. The concept of differential was indeed introduced via an intuitive (or heuristic) definition by Gottfried W. Leibnitz, a German polymath and philosopher of the eighteenth century; its use was widely criticized until Cauchy defined it based on the derivative – which took the central role thereafter, and left dy free for given dx and df / dx as per Eq. (10.1). A graphical representation of differential is conveyed by Fig. 10.1, and the usefulness of differentials to approximate a function becomes clear from inspection thereof; after viewing dy as a small variation in the vertical direction, viz. (10.2) one may retrieve Eq...
