Combining Differentiation Rules
Combining differentiation rules involves using various differentiation techniques, such as the power rule, product rule, quotient rule, and chain rule, to find the derivative of a function. By applying these rules in combination, it becomes possible to differentiate more complex functions by breaking them down into simpler components and applying the appropriate rules to each part.
5 Key excerpts on "Combining Differentiation Rules"
- Stu Schwartz(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Rather than trying to identify the form of a function (such as product or quotient), and then applying the derivative rule, the chain rule is a technique that oversees all the rules of differentiation. The chain rule is always in effect! The chain rule can be expressed several ways and can be confusing to students. Like trying to explain how to ride a bike, sometimes it is easier just to try it. if y = f (x) and u = g (x), EXAMPLE 13: If, find f′ (x) and f″ (x). SOLUTION: f ′(x) = (4 x + 1) 1/2 The differentiation technique requires you to treat the (4 x + 1) 1/2 as an entity and apply the power rule to it. EXAMPLE 14: If, find the slope of the line normal to the graph of y at x = 3. SOLUTION: The expression in brackets is the derivative of using the quotient rule. Slope of normal line at x = 3 is. EXAMPLE 15: The table below gives values of the functions f and g and their derivatives at selected values. of x with a being a constant. If the slope of the tangent line to f (g (x)) at x = 1 is 5, find the value of a. SOLUTION: DID YOU KNOW? There are many different notations for derivatives. The earliest,, was introduced by Gottfried Wilhelm Leibniz (1646−1716). Joseph-Louis Lagrange (1736− 1813) introduced the prime notation such as f′ and f″. However there are several more obscure notations that still survive. Leonhard Euler (1707-1873) used the notation D x y as the first derivative and D 2 x y as the second derivative. Sir Isaac Newton (1642−1727) used dot notation, representing the first two derivatives as and. Tangent Line Approximations Overview : Now that we can calculate the slope of the tangent line by computing the derivative of f and evaluating it at some value of x, we take the next step and actually find the equation of the tangent line. Recall that if we have a graph of y = f (x) passing through a point (x 1, y 1), the equation of the tangent line using the point - slope formula is: y − y 1 = m (x − x 1)...
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- 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,...
