Chapter 2 DERIVATIVES

Transcription

1 Chapter DERIVATIVES Imagine a driver speeding down a highwa, at 0 km/h. He hears a police siren and is quickl pulled over. The police officer tells him that he was speeding, but the driver argues that because he has travelled 00 km from home in two hours, his average speed is within the 00 km/h limit. The driver s argument fails because police officers charge speeders based on their instantaneous speed, not their average speed. There are man other situations in which the instantaneous rate of change is more important than the average rate of change. In calculus, the derivative is a tool for finding instantaneous rates of change. This chapter shows how the derivative can be determined and applied in a great variet of situations. CHAPTER EXPECTATIONS In this chapter, ou will understand and determine derivatives of polnomial and simple rational functions from first principles, Section. identif eamples of functions that are not differentiable, Section. justif and use the rules for determining derivatives, Sections.,.,.,.5 identif composition as two functions applied in succession, Section.5 determine the composition of two functions epressed in notation, and decompose a given composite function into its parts, Section.5 use the derivative to solve problems involving instantaneous rates of change, Sections.,.,.,.5

2 Review of Prerequisite Skills Before beginning our stud of derivatives, it ma be helpful to review the following concepts from previous courses and the previous chapter: Working with the properties of eponents Simplifing radical epressions Finding the slopes of parallel and perpendicular lines Simplifing rational epressions Epanding and factoring algebraic epressions Evaluating epressions Working with the difference quotient Eercise. Use the eponent laws to simplif each of the following epressions. Epress our answers with positive eponents. p 7 6p 9 a. a 5 a c. e. e 6 e p 5 a a b b. a d. a b 5 a 6 b f. a 5 b. Simplif and write each epression in eponential form. Va Va a. b. 8 6 c. Va. Determine the slope of a line that is perpendicular to a line with each given slope. 5 a. b. c. d.. Determine the equation of each of the following lines: a. passing through points A, and B9, b. passing through point A, 5 and parallel to the line 5 c. perpendicular to the line and passing through point A, 6 6 REVIEW OF PREREQUISITE SKILLS

3 5. Epand, and collect like terms. a. d. 5 8 b. e. 5 c. 6 7 f Simplif each epression. a. 5 d. 5 0 b. 5 7 e. 5 5 c. f. h k 9 h k 7. Factor each epression completel. a. k 9 c. a a 7 e. b. d. f. r 5r 8. Use the factor theorem to factor the following epressions: a. a b b. a 5 b 5 c. a 7 b 7 d. a n b n 9. If f 7, evaluate a. f b. f c. f a d. b f Rationalize the denominator in each of the following epressions: V V V V a. b. c. d. V V V V V. a. If f, determine the epression for the difference quotient f a h f a when a. Eplain what this epression can be used for. h b. Evaluate the epression ou found in part a. for a small value of h where h 0.0. c. Eplain what the value ou determined in part b. represents. CHAPTER 6

4 CAREER LINK Investigate CHAPTER : THE ELASTICITY OF DEMAND Have ou ever wondered how businesses set prices for their goods and services? An important idea in marketing is elasticit of demand, or the response of consumers to a change in price. Consumers respond differentl to a change in the price of a staple item, such as bread, than the do to a change in the price of a luur item, such as jeweller. A famil would probabl still bu the same quantit of bread if the price increased b 0%. This is called inelastic demand. If the price of a gold chain, however, increased b 0%, sales would likel decrease 0% or more. This is called elastic demand. Mathematicall, elasticit is defined as the negative of the relative (percent) change in the number demanded Q n n R divided b the relative (percent) change in the price For eample, if a store increased the price of a CD from $7.99 to $9.99, and the number sold per week went from 0 to 80, the elasticit would be E ca An elasticit of about means that the change in demand is three times as large, in percent terms, as the change in price. The CDs have an elastic demand because a small change in price can cause a large change in demand. In general, goods or services with elasticities greater than one E 7 are considered elastic (e.g., new cars), and those with elasticities less than one E 6 are considered inelastic (e.g., milk). In our eample, we calculated the average elasticit between two price levels, but, in realit, businesses want to know the elasticit at a specific, or instantaneous, price level. In this chapter, ou will develop the rules of differentiation that will enable ou to calculate the instantaneous rate of change for several classes of functions. Case Stud Marketer: Product Pricing In addition to developing advertising strategies, marketing departments also conduct research into, and make decisions on, pricing. Suppose that the demand price relationship for weekl movie rentals at a convenience store is np 500 where np is demand and p is price. p, DISCUSSION QUESTIONS E ca n p b a n p bd b a bd Q p p R:. Generate two lists, each with at least five goods and services that ou have purchased recentl, classifing each of the goods and services as having elastic or inelastic demand.. Calculate and discuss the elasticit if a movie rental fee increases from $.99 to $ CAREER LINK

5 Section. The Derivative Function In this chapter, we will etend the concepts of the slope of a tangent and the rate of change to introduce the derivative. We will eamine the methods of differentiation, which we can use to determine the derivatives of polnomial and rational functions. These methods include the use of the power rule, sum and difference rules, and product and quotient rules, as well as the chain rule for the composition of functions. The Derivative at a Point f a h f a In the previous chapter, we encountered limits of the form lim. hs0 h This limit has two interpretations: the slope of the tangent to the graph f at the point a, f a, and the instantaneous rate of change of f with respect to at a. Since this limit plas a central role in calculus, it is given a name and a concise notation. It is called the derivative of f() at a. It is denoted b f a and is read as f prime of a. The derivative of f at the number a is given b provided that this limit eists. f a hs0 f a h f a, h EXAMPLE Selecting a limit strateg to determine the derivative at a number Determine the derivative of f at. Solution Using the definition, the derivative at f h f f h 9 6h h 9 h 6 h 6 h hs0 6 is given b Therefore, the derivative of f at is 6. (Epand) (Simplif and factor) CHAPTER 65

6 An alternative wa of writing the derivative of f at the number a is f f a f a. Sa a In applications where we are required to find the value of the derivative for a number of particular values of, using the definition repeatedl for each value is tedious. The net eample illustrates the efficienc of calculating the derivative of at an arbitrar value of and using the result to determine the derivatives at a number of particular -values. f EXAMPLE Connecting the derivative of a function to an arbitrar value a. Determine the derivative of f at an arbitrar value of. b. Determine the slopes of the tangents to the parabola at, 0, and. Solution a. Using the definition, f h f f h h h h h h hs0 (Epand) (Simplif and factor) The derivative of f at an arbitrar value of is f. b. The required slopes of the tangents to are obtained b evaluating the derivative f at the given -values. We obtain the slopes b substituting for : f f 0 0 f The slopes are, 0, and, respectivel. In fact, knowing the -coordinate of a point on the parabola, we can easil find the slope of the tangent at that point. For eample, given the -coordinates of points on the curve, we can produce the following table. 66. THE DERIVATIVE FUNCTION

7 For the Parabola f The slope of the tangent to the curve f P(, ) -Coordinate of P Slope of Tangent at P at a point P, is given b the derivative f. For each -value, there is an, associated value., 0, 0 0 0,, a, a a a The graphs of f and the derivative function f are shown below. The tangents at, 0, and are shown on the graph of f. 0 f() = 0 (, ) Notice that the graph of the derivative function of the quadratic function (of degree two) is a linear function (of degree one). (, ) f 9() = INVESTIGATION A. Determine the derivative with respect to of each of the following functions: a. f b. f c. f 5 B. In Eample, we showed that the derivative of f is f. Referring to step, what pattern do ou see developing? C. Use the pattern from step to predict the derivative of f 9. D. What do ou think f would be for f n, where n is a positive integer? The Derivative Function The derivative of f at a is a number f a. If we let a be arbitrar and assume a general value in the domain of f, the derivative f is a function. For eample, if f, f, which is itself a function. CHAPTER 67

8 The Definition of the Derivative Function The derivative of f with respect to is the function f, where f h f f, provided that this limit eists. hs0 h The f notation for this limit was developed b Joseph Louis Lagrange (76 8), a French mathematician. When ou use this limit to determine the derivative of a function, it is called determining the derivative from first principles. In Chapter, we discussed velocit at a point. We can now define (instantaneous) velocit as the derivative of position with respect to time. If the position of a bod at time t is st, then the velocit of the bod at time t is vt s t hs0 st h s t. h Likewise, the (instantaneous) rate of change of f with respect to is the f h f function f, whose value is f. hs0 h EXAMPLE Determining the derivative from first principles Determine the derivative f t of the function f t Vt, t 0. Solution Using the definition, f t h f t f t Vt h Vt Vt h Vt t h t Vt h Vt h Vt h Vt hs0 Vt h Vt Vt, for t 7 0 a Vt h Vt Vt h Vt b (Rationalize the numerator) 68. THE DERIVATIVE FUNCTION

9 Note that f t Vt is defined for all instances of t 0, whereas its derivative f t is defined onl for instances when t 7 0. From this, we can see that Vt a function need not have a derivative throughout its entire domain. EXAMPLE Selecting a strateg involving the derivative to determine the equation of a tangent Determine an equation of the tangent to the graph of f at the point where. Solution When, The graph of the., = point Q,, and the tangent at the point are shown. R tangent, First find f. 0 f h f f h hs0 h h h hs0 h h h h (Simplif the fraction) The slope of the tangent at is m f The equation of the tangent is. or, in standard form, 0. CHAPTER 69

10 EXAMPLE 5 Selecting a strateg involving the derivative to solve a problem Determine an equation of the line that is perpendicular to the tangent to the graph of f at and that intersects it at the point of tangenc. Solution In Eample, we found that the slope of the tangent at = is f and, the point of tangenc is Q, The perpendicular line has slope, the negative R. reciprocal of Therefore, the required equation is, or tangent 0 = normal, The line whose equation we found in Eample 5 has a name. The normal to the graph of f at point P is the line that is perpendicular to the tangent at P. The Eistence of Derivatives A function f is said to be differentiable at a if f a eists. At points where f is not differentiable, we sa that the derivative does not eist. Three common was for a derivative to fail to eist are shown. a a a Cusp Vertical Tangent Discontinuit 70. THE DERIVATIVE FUNCTION

11 EXAMPLE 6 Reasoning about differentiabilit at a point Show that the absolute value function f is not differentiable at 0. Solution The graph of f is shown. Because the slope for 6 0 is, whereas the slope for 7 0 is, the graph has a corner at 0, 0, which prevents a unique tangent from being drawn there. We can show this using the definition of a derivative. = 0 f 0 h f 0 f 0 Now, we will consider one-sided limits. 0h 0 h when h 7 0 and 0h 0 hwhen h h 0 lim hs0 h 0h 0 lim hs0 h f h 0 0 h 0 h hs0 h h hs0 h hs0 hs0 Since the left-hand limit and the right-hand limit are not the same, the derivative does not eist at 0. From Eample 6, we conclude that it is possible for a function to be continuous at a point and et not differentiable at this point. However, if a function is differentiable at a point, then it is also continuous at this point. CHAPTER 7

12 Other Notation for Derivatives Smbols other than f are often used to denote the derivative. If f, the d d smbols and are used instead of f. The notation was originall used d d b Leibniz and is read dee b dee. For eample, if, the derivative is d or, in Leibniz notation,. Similarl, in Eample, we showed d that if d then. The Leibniz notation reminds us of the process b, d which the derivative is obtained namel, as the limit of a difference quotient: d d S0 B omitting and f altogether, we can combine these notations and write d which is read the derivative of with respect to is. It is d, d important to note that is not a fraction. d IN SUMMARY Ke Ideas fa h fa The derivative of a function f at a point a, fa is f a, f fa or f a if the limit eists. Sa a A function is said to be differentiable at a if f a eists. A function is differentiable on an interval if it is differentiable at ever number in the interval. The derivative function for an function f is given b f h f f, if the limit eists. Need to Know fa h fa To find the derivative at a point a, ou can use lim. The derivative f a can be interpreted as either the slope of the tangent at a, fa, or the instantaneous rate of change of f with respect to when a. Other notations for the derivative of the function f are f,, d and d. The normal to the graph of a function at point P, is a line that is perpendicular to the tangent line that passes through point P. 7. THE DERIVATIVE FUNCTION

13 Eercise. PART A. State the domain on which f is differentiable. a. d. 0 = f() = f() 0 b. e. = f() 0 = f() 0 c. f. 0 = f() = f() 0 C. Eplain what the derivative of a function represents.. Illustrate two situations in which a function does not have a derivative at.. For each function, find f a h and f a h f a. a. f 5 d. f 6 b. f e. f 7 c. f f. f CHAPTER 7

14 K PART B 5. For each function, find the value of the derivative f a for the given value of a. a. f, a c. f V, a 0 b. f, a d. f 5, a 6. Use the definition of the derivative to find f for each function. a. f 5 8 c. f 6 7 b. f d. f V d 7. In each case, find the derivative from first principles. d a. 6 7 b. c. 8. Determine the slope of the tangents to when 0,, and. Sketch the graph, showing these tangents. A 9. a. Sketch the graph of f. b. Calculate the slopes of the tangents to f at points with -coordinates,, 0,,. c. Sketch the graph of the derivative function f. d. Compare the graphs of f and f. 0. An object moves in a straight line with its position at time t seconds given b st t 8t, where s is measured in metres. Find the velocit when t 0, t, and t 6.. Determine an equation of the line that is tangent to the graph of f V and parallel to 6 0. d. For each function, use the definition of the derivative to determine where d, a, b, c, and m are constants. a. c c. m b b. d. a b c. Does the function f ever have a negative slope? If so, where? Give reasons for our answer.. A football is kicked up into the air. Its height, h, above the ground, in metres, at t seconds can be modelled b ht 8t.9t. a. Determine h. b. What does h represent? 7. THE DERIVATIVE FUNCTION

15 T 5. Match each function in graphs a, b, and c with its corresponding derivative, graphed in d, e, and f. a. d. 0 0 b. e. 0 0 c. f. 0 0 PART C 6. For the function f 0 0, show that f 0 eists. What is the value? 7. If f a 0 and f a 6, find lim hs0 f a h. h 8. Give an eample of a function that is continuous on q 6 6qbut is not differentiable at. 9. At what point on the graph of 5 is the tangent parallel to? 0. Determine the equations of both lines that are tangent to the graph of f and pass through point,. CHAPTER 75

16 Section. The Derivatives of Polnomial Functions We have seen that derivatives of functions are of practical use because the represent instantaneous rates of change. Computing derivatives from the limit definition, as we did in Section., is tedious and time-consuming. In this section, we will develop some rules that simplif the process of differentiation. We will begin developing the rules of differentiation b looking at the constant function, f k. Since the graph of an constant function is a horizontal line with slope zero at each point, the derivative should be zero. For eample, d if f, then f 0. Alternativel, we can write 0. d 0 f() = slope = 0 (, ) The Constant Function Rule If f k, where k is a constant, then f 0. In Leibniz notation, d d k 0. Proof: f hs0 f h f h k k 0 hs0 0 (Since f k and f h k for all h) 76. THE DERIVATIVES OF POLYNOMIAL FUNCTIONS

17 EXAMPLE Appling the constant function rule a. If f 5, f 0. b. If p, d 0. d A power function is a function of the form f n, where n is a real number. In the previous section, we observed that for f, f ; for g g ; and for h, h. V, d As well, we hpothesized that In fact, this is true and is called d n n n. the power rule. The Power Rule If f n, where n is a real number, then f n n. In Leibniz notation, d d n n n. Proof: (Note: n is a positive integer.) Using the definition of the derivative, f hs0 f h f, where f n h h n n h h n h n p hs0 h h n h n p h n n hs0 n n p n n n n p n n n n h n n h (Factor) (Divide out h) (Since there are n terms) CHAPTER 77

18 EXAMPLE Appling the power rule a. If f 7, then f 7 6. b. If g then g,. ds c. If s t, dt t Vt. d d. d 0 The Constant Multiple Rule If f kg, where k is a constant, then f kg. In Leibniz notation, d k kd d d. Proof: Let f kg. B the definition of the derivative, f h f f kg h kg g h g k c d g h g k lim c d kg (Factor) (Propert of limits) EXAMPLE Appling the constant multiple rule Differentiate the following functions: a. f 7 b. Solution a. f 7 b. f 7 d d 7 d d d d a b THE DERIVATIVES OF POLYNOMIAL FUNCTIONS

19 We conclude this section with the sum and difference rules. The Sum Rule If functions p and q are differentiable, and f p q, then f p q. d In Leibniz notation, f d p d q. d d d Proof: Let f p q. B the definition of the derivative, f h f f p h q h p q hs0 h p h p e p h p e p q q h q f h q h q f e f The Difference Rule If functions p and q are differentiable, and f p q, then f p q. d In Leibniz notation, f d p d q. d d d The proof for the difference rule is similar to the proof for the sum rule. EXAMPLE Selecting appropriate rules to determine the derivative Differentiate the following functions: a. f 5V b. CHAPTER 79

20 Solution We appl the constant multiple, power, sum, and difference rules. a. f 5V b. We first epand. f d d d d 5 9 d d 5 d d 5 a b d 9 0 d 8 6 5, or 6 5 V EXAMPLE 5 Selecting a strateg to determine the equation of a tangent Determine the equation of the tangent to the graph of f at. Solution A Using the derivative The slope of the tangent to the graph of f at an point is given b the derivative f. For f f 6 Now, f 6 6 The slope of the tangent at is and the point of tangenc is, f, 0. The equation of the tangent is 0 or. Tech Support For help using the graphing calculator to graph functions and draw tangent lines See Technical Appendices p. 597 and p Solution B Using the graphing calculator Draw the graph of the function using the graphing calculator. Draw the tangent at the point on the function where. The calculator displas the equation of the tangent line. The equation of the tangent line in this case is. 80. THE DERIVATIVES OF POLYNOMIAL FUNCTIONS

21 EXAMPLE 6 Connecting the derivative to horizontal tangents Determine points on the graph in Eample 5 where the tangents are horizontal. Solution Horizontal lines have slope zero. We need to find the values of that satisf f or The graph of f has horizontal tangents at 0, and,. IN SUMMARY Ke Ideas The following table summarizes the derivative rules in this section. Rule Function Notation Leibniz Notation Constant Function Rule If f k, where k is a constant, then f 0. d k 0 d Power Rule If f n, where n is a real number, then f n n. Constant Multiple Rule If f kg, then f kg. Sum Rule If f p q, then f p q. Difference Rule If f p q, then f p q. d d n n n d k kd d d d d f d d p d d q d d f d d p d d q Need to Know To determine the derivative of a simple rational function, such as f, 6 epress the function as a power, then use the power rule. If f 6, then f If ou have a radical function such as g 5, rewrite the function as 5 g, then use the power rule. 5 If, then 5 g 5 g CHAPTER 8

22 Eercise. PART A. What rules do ou know for calculating derivatives? Give eamples of each rule.. Determine f for each of the following functions: a. f 7 c. f 5 8 e. b. f d. f V f. f a b f K. Differentiate each function. Use either Leibniz notation or prime notation, depending on which is appropriate. a. h d. 5 5 b. f 5.75 e. c. s t t t f. g 5. Appl the differentiation rules ou learned in this section to find the derivatives of the following functions: a. 5 c. 6 e. st t5 t, t 7 0 t V 6V V 6 b. d. 9 V f. V PART B 5. Let s represent the position of a moving object at time t. Find the velocit v ds at time t. dt a. s t 7t b. s 8 5t c. s t t 6. Determine f a for the given function f at the given value of a. a. f V, a b. f 7 6V 5, a 6 7. Determine the slope of the tangent to each of the curves at the given point. a.,, c.,, b.,, d. V6,, 5 8. THE DERIVATIVES OF POLYNOMIAL FUNCTIONS

23 8. Determine the slope of the tangent to the graph of each function at the point with the given -coordinate. a., c. 6, b. V 5, d., 9. Write an equation of the tangent to each curve at the given point. a., P0.5, d. a b, P, C T A b. e., P, 7 c. V, P, 9 f. 0. What is a normal to the graph of a function? Determine the equation of the normal to the graph of the function in question 9, part b., at the given point.. Determine the values of so that the tangent to the function is parallel V to the line Do the functions and ever have the same slope? If so, where?. Tangents are drawn to the parabola at, and Q. Prove that 8, 6 R these lines are perpendicular. Illustrate with a sketch.. Determine the point on the parabola where the slope of the tangent is 5. Illustrate our answer with a sketch. 5. Determine the coordinates of the points on the graph of at which the slope of the tangent is. 6. Show that there are two tangents to the curve that have a slope of Determine the equations of the tangents to the curve that pass through the following points: a. point, b. point, 7 8. Determine the value of a, given that the line a 0 is tangent to the graph of a at. 9. It can be shown that, from a height of h metres, a person can see a distance of d kilometres to the horizon, where d.5vh. a. When the elevator of the CN Tower passes the 00 m height, how far can the passengers in the elevator see across Lake Ontario? b. Find the rate of change of this distance with respect to height when the height of the elevator is 00 m. V V 8, P, 0 V, P, V CHAPTER 8

24 0. An object drops from a cliff that is 50 m high. The distance, d, in metres, that the object has dropped at t seconds in modelled b dt.9t. a. Find the average rate of change of distance with respect to time from s to 5 s. b. Find the instantaneous rate of change of distance with respect to time at s. c. Find the rate at which the object hits the ground to the nearest tenth.. A subwa train travels from one station to the net in min. Its distance, in kilometres, from the first station after t minutes is st t At what t. times will the train have a velocit of 0.5 km>min?. While working on a high-rise building, a construction worker drops a bolt from 0 m above the ground. After t seconds, the bolt has fallen a distance of s metres, where st 5t, 0 t 8. The function that gives the height of the bolt above ground at time t is Rt 0 5t. Use this function to determine the velocit of the bolt at t.. Tangents are drawn from the point 0, to the parabola. Find the coordinates of the points at which these tangents touch the curve. Illustrate our answer with a sketch.. The tangent to the cubic function that is defined b 6 8 at point A, intersects the curve at another point, B. Find the coordinates of point B. Illustrate with a sketch. 5. a. Find the coordinates of the points, if an, where each function has a horizontal tangent line. i. f 5 ii. f iii. f 8 5 b. Suggest a graphical interpretation for each of these points. PART C 6. Let Pa, b be a point on the curve V V. Show that the slope b of the tangent at P is Å. a 7. For the power function f n, find the -intercept of the tangent to its graph at point,. What happens to the -intercept as n increases without bound n S q? Eplain the result geometricall. 8. For each function, sketch the graph of f and find an epression for f. Indicate an points at which f does not eist., 6 a. f e b. f c. f , 8. THE DERIVATIVES OF POLYNOMIAL FUNCTIONS

25 Section. The Product Rule In this section, we will develop a rule for differentiating the product of two functions, such as f 8 and g, without first epanding the epressions. You might suspect that the derivative of a product of two functions is simpl the product of the separate derivatives. An eample shows that this is not so. EXAMPLE Reasoning about the derivative of a product of two functions Let p f g, where f and g 5. Show that p f g. Solution The epression p can be simplified. p p 0 f and g, so f g. Since is not the derivative of p, we have shown that p f g. The correct method for differentiating a product of two functions uses the following rule. The Product Rule If p f g, then p f g f g. If u and v are functions of, d du dv uv v u d d d. In words, the product rule sas, the derivative of the product of two functions is equal to the derivative of the first function times the second function plus the first function times the derivative of the second function. Proof: p f g, so using the definition of the derivative, p hs0 f hg h f g. h CHAPTER 85

26 To evaluate p, we subtract and add the same term in the numerator. Now, f hg h f g h f g h f g p hs0 h f h f g h g ec d g h f c df h f h f c f g f g d limg h hs0 hs0 f lim c hs0 g h g d h EXAMPLE Appling the product rule Differentiate h 5 Solution h 5 Using the product rule, we get using the product rule. h d d d d 5 We can, of course, differentiate the function after we first epand. The product rule will be essential, however, when we work with products of polnomials such as f 9 5 or non-polnomial functions such as f 9 V 5. It is not necessar to simplif an epression when ou are asked to calculate the derivative at a particular value of. Because man epressions obtained using differentiation rules are cumbersome, it is easier to substitute, then evaluate the derivative epression. The net eample could be solved b finding the product of the two polnomials and then calculating the derivative of the resulting polnomial at. Instead, we will appl the product rule. 86. THE PRODUCT RULE

27 EXAMPLE Selecting an efficient strateg to determine the value of the derivative Find the value h for the function h Solution h Using the product rule, we get h h The following eample illustrates the etension of the product rule to more than two functions. EXAMPLE Connecting the product rule to a more comple function Find an epression for p if p f gh. Solution We temporaril regard f g as a single function. p f gh B the product rule, p f g h f gh A second application of the product rule ields p f g f g h f gh f gh f g h f gh This epression gives us the etended product rule for the derivative of a product of three functions. Its smmetrical form makes it eas to etend to a product of four or more functions. The Power of a Function Rule for Positive Integers Suppose that we now wish to differentiate functions such as or 5 6. CHAPTER 87

28 These functions are of the form u n, where n is a positive integer and u g is a function whose derivative we can find. Using the product rule, we can develop an efficient method for differentiating such functions. For n, h g h gg Using the product rule, h g g gg g g Similarl, for n, we can use the etended product rule. Thus, h g ggg h g gg gg g ggg g g These results suggest a generalization of the power rule. The Power of a Function Rule for Integers If u is a function of, and n d is an integer, then d un nu n du d. In function notation, if f g n, then f ng n g. The power of a function rule is a special case of the chain rule, which we will discuss later in this chapter. We are now able to differentiate an polnomial, such as h 5 6 or h 6, without multipling out 5 the brackets. We can also differentiate rational functions, such as f. EXAMPLE 5 Appling the power of a function rule For h 5 6, find h. Solution Here h has the form h g 6, where the inner function is g 5. B the power of a function rule, we get h THE PRODUCT RULE

29 EXAMPLE 6 Selecting a strateg to determine the derivative of a rational function 5 Differentiate the rational function f b first epressing it as a product and then using the product rule. Solution 5 f 5 (Epress f as a product) f d d 5 5 d d (Product rule) 5 d d (Power of a function rule) (Simplif) EXAMPLE 7 Using the derivative to solve a problem The position s, in centimetres, of an object moving in a straight line is given b s t6 t, t 0, where t is the time in seconds. Determine the object s velocit at t. Solution The velocit of the object at an time v d dt t6 t t 0 is v ds dt. 6 t t d dt 6 t 6 t t6 t At t, v We conclude that the object is at rest at t s. (Product rule) (Power of a function rule) CHAPTER 89

30 IN SUMMARY Ke Ideas The derivative of a product of differentiable functions is not the product of their derivatives. The product rule for differentiation: If h fg, then h f g fg. The power of a function rule for integers: If f g n, then f ng n g. Need to Know In some cases, it is easier to epand and simplif the product before differentiating, rather than using the product rule. If f f If the derivative is needed at a particular value of the independent variable, it is not necessar to simplif before substituting. Eercise. K PART A. Use the product rule to differentiate each function. Simplif our answers. a. h d. h 5 7 b. h e. st t t c. h 7 f. f. Use the product rule and the power of a function rule to differentiate the following functions. Do not simplif. a. 5 c. 6 b. 5 d. 9. When is it not appropriate to use the product rule? Give eamples.. Let F bc. Epress F in terms of b and c. 90. THE PRODUCT RULE

31 PART B d 5. Determine the value of for the given value of. d a. 7, b. c. d. e. f.,, 7, 5, 5 5, 6. Determine the equation of the tangent to the curve 5 at the point,. 7. Determine the point(s) where the tangent to the curve is horizontal. a. 9 b. 8. Use the etended product rule to differentiate the following functions. Do not simplif. a. b. A C T 9. A 75 L gas tank has a leak. After t hours, the remaining volume, V, in litres is Vt 75 a t Use the product rule to determine how b, 0 t. quickl the gas is leaking from the tank when the tank is 60% full of gas. 0. Determine the slope of the tangent to h at. Eplain how to find the equation of the normal at. PART C. a. Determine an epression for f if f g g g p g n g n. b. If f p n, find f 0.. Determine a quadratic function f a b c if its graph passes through the point, 9 and it has a horizontal tangent at, 8.. Sketch the graph of f 0 0. a. For what values of is f not differentiable? b. Find a formula for f, and sketch the graph of f. c. Find f () at, 0, and.. Show that the line 0 is tangent to the curve 6. CHAPTER 9

32 Mid-Chapter Review. a. Sketch the graph of f 5. b. Calculate the slopes of the tangents to f 5 at points with -coordinates 0,,,..., 5. c. Sketch the graph of the derivative function f. d. Compare the graphs of f and f.. Use the definition of the derivative to find f for each function. a. f 6 5 c. f 5 5 b. f d. f V. a. Determine the equation of the tangent to the curve at. b. Sketch the graph of the function and the tangent.. Differentiate each of the following functions: a. 6 c. g e. t b. 0 d. 5 f. 5. Determine the equation of the tangent to the graph of f that has slope. 6. Determine f for each of the following functions: a. f 7 8 d. f V V b. f 5 6 e. f 7 V c. f 5 f. f 5 7. Determine the equation of the tangent to the graph of each function. a. 6 when b. V when 9 c. f 8 9 when 8. Determine the derivative using the product rule. a. f 9 5 c. 6 9 b. f t t 7t 8t d. 9 MID-CHAPTER REVIEW

33 9. Determine the equation of the tangent to 5 9 at, Determine the point(s) where the tangent to the curve 5 is horizontal. d. If 5 8, determine from first principles. d. A tank holds 500 L of liquid, which takes 90 min to drain from a hole in the bottom of the tank. The volume, V, remaining in the tank after t minutes is Vt 500 a t 90 b, where 0 t 90 a. How much liquid remains in the tank at h? b. What is the average rate of change of volume with respect to time from 0 min to 60 min? c. How fast is the liquid draining at 0 min?. The volume of a sphere is given b Vr. pr a. Determine the average rate of change of volume with respect to radius as the radius changes from 0 cm to 5 cm. b. Determine the rate of change of volume when the radius is 8 cm.. A classmate sas, The derivative of a cubic polnomial function is a quadratic polnomial function. Is the statement alwas true, sometimes true, or never true? Defend our choice in words, and provide two eamples to support our argument. d 5. Show that and a and b are integers. d a ba b if a b a b 6. a. Determine f, where f b. Give two interpretations of the meaning of f. 7. The population, P, of a bacteria colon at t hours can be modelled b Pt 00 0t 0t t a. What is the initial population of the bacteria colon? b. What is the population of the colon at 5 h? c. What is the growth rate of the colon at 5 h? 8. The relative percent of carbon dioide, C, in a carbonated soft drink at t minutes can be modelled b Ct 00, where t 7. Determine C t t and interpret the results at 5 min, 50 min, and 00 min. Eplain what is happening. CHAPTER 9

34 Section. The Quotient Rule In the previous section, we found that the derivative of the product of two functions is not the product of their derivatives. The quotient rule gives the derivative of a function that is the quotient of two functions. It is derived from the product rule. The Quotient Rule If h f then h g, f g f g g, du dv In Leibniz notation, d Q u d v R v u d d v. g 0. Proof: We want to find h, given that h f, g 0. g We rewrite this as a product: hg f. Using the product rule, h g hg f. f hg Solving for h, we get h g f f g g g f g f g g The quotient rule provides us with an alternative approach to differentiate rational functions, in addition to what we learned last section. Memor Aid for the Product and Quotient Rules It is worth noting that the quotient rule is similar to the product rule in that both have f g and f g. For the product rule, we put an addition sign between the terms. For the quotient rule, we put a subtraction sign between the terms and then divide the result b the square of the original denominator. Take note that in the quotient rule the f g the case with the product rule. term must come first. This isn't 9. THE QUOTIENT RULE

35 EXAMPLE Appling the quotient rule Determine the derivative of h 5. Solution Since h f where f and g 5, use the quotient g, rule to find h. f g f g Using the quotient rule, we get h g EXAMPLE Tech Support For help using the graphing calculator to graph functions and draw tangent lines see Technical Appendices p. 597 and p Selecting a strateg to determine the equation of a line tangent to a rational function Determine the equation of the tangent to at 0. Solution A Using the derivative d The slope of the tangent to the graph of f at an point is given b the derivative d. B the quotient rule, d d At 0, d 0 00 d 0 The slope of the tangent at 0 is and the point of tangenc is 0, 0. The equation of the tangent is. Solution B Using the graphing calculator Draw the graph of the function using the graphing calculator. Draw the tangent at the point on the function where 0. The calculator displas the equation of the tangent line. The equation of the tangent line in this case is. CHAPTER 95

36 EXAMPLE Using the quotient rule to solve a problem 8 Determine the coordinates of each point on the graph of f where the tangent is horizontal. V Solution The slope of the tangent at an point on the graph is given b f. Using the quotient rule, V 8a b f V 8 V V V V V The tangent will be horizontal when f 0; that is, when. The point on the graph where the tangent is horizontal is, 8. IN SUMMARY Ke Ideas The derivative of a quotient of two differentiable functions is not the quotient of their derivatives. The quotient rule for differentiation: If h f g, then h f g f g g, g 0. Need to Know To find the derivative of a rational function, ou can use two methods: Leave the function in fraction form, OR Epress the function as a product, and use the quotient rule. f and use the product and power of a function rules. f 96. THE QUOTIENT RULE

37 Eercise. PART A. What are the eponent rules? Give eamples of each rule.. Cop the table, and complete it without using the quotient rule. Function Rewrite Differentiate and Simplif, if Necessar f, 0 g 5, 0 h 05, 0 8 6, 0 s t 9 t, t C K T. What are the different was to find the derivative of a rational function? Give eamples. PART B. Use the quotient rule to differentiate each function. Simplif our answers. a. 5 h c. h e. b. t d. f. ht h t 5 d 5. Determine at the given value of. d a. 5, c. 5 5, b. 9, d., 6. Determine the slope of the tangent to the curve at point, Determine the points on the graph of where the slope of the tangent is Show that there are no tangents to the graph of f that have a negative slope. CHAPTER 97

38 A 9. Find the point(s) at which the tangent to the curve is horizontal. a. b. 0. An initial population, p, of 000 bacteria grows in number according to the equation pt 000Q t where t is in hours. Find the rate at t 50 R, which the population is growing after h and after h.. Determine the equation of the tangent to the curve at.. A motorboat coasts toward a dock with its engine off. Its distance s, in metres, from the dock t seconds after the engine is turned off is st 0 6 t for 0 t 6. t a. How far is the boat from the dock initiall? b. Find the velocit of the boat when it bumps into the dock.. a. The radius of a circular juice blot on a piece of paper towel t seconds after it t was first seen is modelled b rt where r is measured in t, centimetres. Calculate i. the radius of the blot when it was first observed ii. the time at which the radius of the blot was.5 cm iii. the rate of increase of the area of the blot when the radius was.5 cm b. According to this model, will the radius of the blot ever reach cm? Eplain our answer. a b. The graph of f has a horizontal tangent line at,. Find a and b. Check using a graphing calculator. 5. The concentration, c, of a drug in the blood t hours after the drug is taken 5t orall is given b ct. When does the concentration reach its t 7 maimum value? 6. The position from its starting point, s, of an object that moves in a straight line at time t seconds is given b st t Determine when the object t 8. changes direction. PART C a b 7. Consider the function f where a, b, c, and d are c d, d c, nonzero constants. What condition on a, b, c, and d ensures that each tangent to the graph of f has a positive slope? 98. THE QUOTIENT RULE

39 Section.5 The Derivatives of Composite Functions Recall that one wa of combining functions is through a process called composition. We start with a number in the domain of g, find its image g, and then take the value of f at g, provided that g is in the domain of f. The result is the new function h f g, which is called the composite function of f and g, and is denoted f g. Definition of a composite function Given two functions f and g, the composite function f g f g f g. is defined b EXAMPLE Reflecting on the process of composition If f V and g 5, find each of the following values: a. f g b. g f c. f g d. g f Solution a. Since g 9, we have f g f 9. b. Since f, we have g f g 7. Note: f g g f. c. f g f 5 V 5 d. g f g V V 5 Note: f g g f. The chain rule states how to compute the derivative of the composite function h f g in terms of the derivatives of f and g. The Chain Rule If f and g are functions that have derivatives, then the composite function h f g has a derivative given b h f gg. In words, the chain rule sas, the derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function. CHAPTER 99

40 Proof: f g h f g B the definition of the derivative, f g. hs0 h Assuming that g h g 0, we can write f g h f g g h g f g ca ba bd hs0 g h g h f g h f g c hs0 g h g Therefore, f g f gg. g h g d lim c d Since lim g h g 0, let g h g k and k S 0 hs0 as h S 0. We obtain f g k f g h g f g c d lim cg ks0 k (Propert of limits) This proof is not valid for all circumstances. When dividing b g h g, we assume that g h g 0. A proof that covers all cases can be found in advanced calculus tetbooks. d EXAMPLE Appling the chain rule Differentiate h. Solution The inner function is g, and the outer function is f. The derivative of the inner function is g. The derivative of the outer function is f. The derivative of the outer function evaluated at the inner function g f. is B the chain rule, h. The Chain Rule in Leibniz Notation If is a function of u and u is a function of (so that is a composite d d du d du function), then provided that and eist. d du d, du d If we interpret derivatives as rates of change, the chain rule states that if is a function of through the intermediate variable u, then the rate of change of 00.5 THE DERIVATIVES OF COMPOSITE FUNCTIONS

41 with respect to is equal to the product of the rate of change of with respect to u and the rate of change of u with respect to. EXAMPLE Appling the chain rule using Leibniz notation If u u, where u V, find d at. d Solution Using the chain rule, d d du d du d u ca bd u a V b It is not necessar to write the derivative entirel in terms of. d When, u V and d Q R 6Q R. EXAMPLE Selecting a strateg involving the chain rule to solve a problem An environmental stud of a certain suburban communit suggests that the average dail level of carbon monoide in the air can be modelled b the function C p V0.5p 7, where C p is in parts per million and population p is epressed in thousands. It is estimated that t ears from now, the population of the communit will be pt. 0.t thousand. At what rate will the carbon monoide level be changing with respect to time three ears from now? Solution dc We are asked to find the value of when t. dt, We can find the rate of change b using the chain rule. Therefore, dc dt dc dp dp dt d0.5p 7 dp d. 0.t c 0.5p 7 0.5pd0.t When t, p. 0.. dc So, dt c d dc Since the sign of is positive, the carbon monoide level will be increasing dt at the rate of 0. parts per million per ear three ears from now. dt CHAPTER 0

42 EXAMPLE 5 Using the chain rule to differentiate a power of a function If 5 7, find d d. Solution The inner function is g 5, and the outer function is f 7. B the chain rule, d d Eample 5 is a special case of the chain rule in which the outer function is a power function of the form g n. This leads to a generalization of the power rule seen earlier. Power of a Function Rule d If n is a real number and then, d un n du u g, nu d or d d gn n g n g. EXAMPLE 6 Tech Support For help using the graphing calculator to graph functions and draw tangent lines see Technical Appendices p. 597 and p Connecting the derivative to the slope of a tangent Using a graphing calculator, sketch the graph of the function Find the equation of the tangent at the point, on the graph. Solution Using a graphing calculator, the graph is The slope of the tangent at point, is given b f. We first write the function as f 8. B the power of a function rule, f 8. The slope at, is f f 8. The equation of the tangent is, or THE DERIVATIVES OF COMPOSITE FUNCTIONS

43 EXAMPLE 7 Combining derivative rules to differentiate a comple product Differentiate h 5. Epress our answer in a simplified factored form. Solution Here we use the product rule and the chain rule. h d d 5 d d (Product rule) (Chain rule) (Simplif) (Factor) EXAMPLE 8 Combining derivative rules to differentiate a comple quotient Determine the derivative of g Q R 0. Solution A Using the product and chain rule There are several approaches to this problem. You could keep the function as it is and use the chain rule and the quotient rule. You could also decompose the function and epress it as g 0, and then appl the quotient rule 0 and the chain rule. Here we will epress the function as the product g 0 0 and appl the product rule and the chain rule. g d d c 0 d 0 0 d d c 0 d (Simplif) (Factor) (Rewrite using positive eponents) Solution B Using the chain and quotient rule In this solution, we will use the chain rule and the quotient rule, where u is the inner function and u is the outer function. 0 g dg du du d CHAPTER 0

44 g d 0 d 9 0 a b d a d b 9 0 a b c d 9 0 a b c d 9 0 a b c d d a d b (Chain rule and quotient rule) (Epand) (Simplif) IN SUMMARY Ke Idea The chain rule: If is a function of u, and u is a function of (i.e., is a composite function), d d du d du then provided that and eist. d du d, du d Therefore, if h f g, then h f g g (Function notation) or dh d df g dg dg d (Leibniz notation) Need to Know When the outer function is a power function of the form g n, we have a special case of the chain rule, called the power of a function rule: d d gn dgn dg dg d ng n g 0.5 THE DERIVATIVES OF COMPOSITE FUNCTIONS

45 Eercise.5 PART A. Given f V and g, find the following value: a. f g c. g f 0 e. f g b. g f d. f g f. g f. For each of the following pairs of functions, find the composite functions f g and g f. What is the domain of each composite function? Are the composite functions equal? a. f b. f c. f C K g V g g V. What is the rule for calculating the derivative of the composition of two differentiable functions? Give eamples, and show how the derivative is determined.. Differentiate each function. Do not epand an epression before differentiating. a. f d. f p b. g e. V c. h 5 f. f 6 5 PART B 5. Rewrite each of the following in the form u n or ku n, and then differentiate. a. c. e. 5 b. d. f Given h g f, where f and g are continuous functions, use the information in the table to evaluate h and h. f() g() f () g () Given f, g, and h f g, determine h. CHAPTER 05

46 8. Differentiate each function. Epress our answer in a simplified factored form. a. f 6 d. h 5 b. e. c. f. a b 9. Find the rate of change of each function at the given value of t. Leave our answers as rational numbers, or in terms of roots and the number p. a. b. s(t) = a t p t p t 6p b st t t 5, t 8, 0. For what values of do the curves and 6 have the same slope?. Find the slope of the tangent to the curve at Q, R.. Find the equation of the tangent to the curve 7 5 at. d. Use the chain rule, in Leibniz notation, to find at the given value of. d a. b. c. d. u 5u, u, u u, u, uu, u, u 5u 7u, u V,. Find h, given h f g, f u u, g, and g. A 5. A L tank can be drained in 0 min. The volume of water remaining in the tank after t minutes is Vt Q t At what rate, 0 R, 0 t 0. to the nearest whole number, is the water flowing out of the tank when t 0? 6. The function st t t, t 0, represents the displacement s, in metres, of a particle moving along a straight line after t seconds. Determine the velocit when t. T PART C 7. a. Write an epression for h if h pqr. b. If h 7, find h. 8. Show that the tangent to the curve at the point, is also the tangent to the curve at another point. 9. Differentiate THE DERIVATIVES OF COMPOSITE FUNCTIONS

47 Technolog Etension: Derivatives on Graphing Calculators Numerical derivatives can be approimated on a TI-8/8 Plus using nderiv(. To approimate f 0 for f follow these steps: Press MATH, and scroll down to 8:nDeriv( under the MATH menu. Press ENTER, and the displa on the screen will be nderiv(. To find the derivative, ke in the epression, the variable, the value at which we want the derivative, and a value for e. For this eample, the displa will be nderiv X>X, X, 0, 0.0. Press ENTER, and the value will be returned. Therefore, f 0is approimatel A better approimation can be found b using a smaller value for e, such as e The default value for e is Tr These: a. Use the nderiv( function on a graphing calculator to determine the value of the derivative of each of the following functions at the given point. i. f, ii. f, iii. f 6, iv. f, 0 v. f 6 V, vi. f, b. Determine the actual value of each derivative at the given point using the rules of differentiation. The TI-89, TI-9, an TI-Nspire can find eact smbolic and numerical derivatives. If ou have access to either model, tr some of the functions above and compare our answers with those found using a TI-8/8 Plus. For eample, on the TI-89 press under the CALCULATE menu, ke d>, 0 0 and ENTER press. CHAPTER 07

48 CAREER LINK WRAP-UP Investigate and Appl CHAPTER : THE ELASTICITY OF DEMAND An electronics retailing chain has established the monthl price p demand n relationship for an electronic game as np p p The are tring to set a price level that will provide maimum revenue (R). The know that when demand is elastic E 7, a drop in price will result in higher overall revenues R np, and that when demand is inelastic E 6, an increase in price will result in higher overall revenues. To complete the questions in this task, ou will have to use the elasticit definition E ca n p b a n p bd converted into differential Q n dn notation. p dp R a. Determine the elasticit of demand at $0 and $80, classifing these price points as having elastic or inelastic demand. What does this sa about where the optimum price is in terms of generating the maimum revenue? Eplain. Also calculate the revenue at the $0 and $80 price points. b. Approimate the demand curve as a linear function (tangent) at a price point of $50. Plot the demand function and its linear approimation on the graphing calculator. What do ou notice? Eplain this b looking at the demand function. c. Use our linear approimation to determine the price point that will generate the maimum revenue. (Hint: Think about the specific value of E where ou will not want to increase or decrease the price to generate higher revenues.) What revenue is generated at this price point? d. A second game has a price demand relationship of 500 n p p 5 The price is currentl set at $50. Should the compan increase or decrease the price? Eplain. 08 CAREER LINK WRAP-UP