Intro diffcall3b
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This document provides an introduction to differential calculus. It begins by motivating the concept of instantaneous velocity and derivative using the example of calculating speed from position over time. It then discusses how the derivative can be used to calculate rates of change in various contexts from mechanics to economics. The document proceeds to define the gradient of tangents and secants to a graph, and introduces methods for calculating derivatives, such as the definition of the derivative and differentiation rules. It concludes by discussing the history of calculus and links to further topics.
Intro diffcall3b
- 1. A guide for teachers – Years 11 and 12 1 2 3 4 5 6 7 8 9 10 11 12 Supporting Australian Mathematics Project Calculus: Module 10 Introduction to differential calculus
- 2. Full bibliographic details are available from Education Services Australia. Published by Education Services Australia PO Box 177 Carlton South Vic 3053 Australia Tel: (03) 9207 9600 Fax: (03) 9910 9800 Email: info@esa.edu.au Website: www.esa.edu.au © 2013 Education Services Australia Ltd, except where indicated otherwise. You may copy, distribute and adapt this material free of charge for non-commercial educational purposes, provided you retain all copyright notices and acknowledgements. This publication is funded by the Australian Government Department of Education, Employment and Workplace Relations. Supporting Australian Mathematics Project Australian Mathematical Sciences Institute Building 161 The University of Melbourne VIC 3010 Email: enquiries@amsi.org.au Website: www.amsi.org.au Editor: Dr Jane Pitkethly, La Trobe University Illustrations and web design: Catherine Tan, Michael Shaw Introduction to differential calculus – A guide for teachers (Years 11–12) Principal author: Dr Daniel Mathews, Monash University Peter Brown, University of NSW Dr Michael Evans, AMSI Associate Professor David Hunt, University of NSW
- 3. Assumed knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 How fast are you really going? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The more things change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The gradient of secants and tangents to a graph . . . . . . . . . . . . . . . . . . . 7 Calculating the gradient of y = x2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Definition of the derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Notation for the derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Some derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Properties of the derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 The product, quotient and chain rules . . . . . . . . . . . . . . . . . . . . . . . . . 19 Summary of differentiation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 The tangent line to a graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 The second derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Differentiation of inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Implicit differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Links forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Higher derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Approximations of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 History and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 The discoverers of calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 The Newton–Leibniz controversy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Mathematically rigorous calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Functions differentiable and not . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Answers to exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
- 4. Introduction to differential calculus Assumed knowledge The content of the modules: • Coordinate geometry • The binomial theorem • Functions I • Functions II • Limits and continuity. Motivation How fast are you really going? You ride your bicycle down a straight bike path. As you proceed, you keep track of your exact position — so that, at every instant, you know exactly where you are. How fast were you going one second after you started? This is not a very realistic situation — it’s hard to measure your position precisely at every instant! Nonetheless, let us suspend disbelief and imagine you have this information. (Perhaps you have an extremely accurate GPS or a high-speed camera.) By considering this question, we are led to some important mathematical ideas. First, recall the formula for average velocity: v = ∆x ∆t , where ∆x is the change in your position and ∆t is the time taken. Thus, average velocity is the rate of change of position with respect to time.
- 5. A guide for teachers – Years 11 and 12 • {5} Draw a graph of your position x(t) at time t seconds. Connect two points on the graph, representing your position at two different times. The gradient of this line is your average velocity over that time period. x t 0 ∆t ∆x Average velocity v = ∆x ∆t . Trying to discover your velocity at the one-second mark (t = 1), you calculate your aver- age velocity over the period from t = 1 to a slightly later time t = 1+∆t. Trying to be more accurate, you look at shorter time intervals, with ∆t smaller and smaller. If you really knew your position at every single instant of time, then you could work out your aver- age velocity over any time interval, no matter how short. The three lines in the following diagram correspond to ∆t = 2, ∆t = 1 and ∆t = 0.5. x t 0 1 1.5 2 3 Average velocity over shorter time intervals. As ∆t approaches 0, you obtain better and better estimates of your instantaneous veloc- ity at the instant t = 1. These estimates correspond to the gradients of lines connecting closer and closer points on the graph.
- 6. {6} • Introduction to differential calculus In the limit, as ∆t → 0, lim ∆t→0 ∆x ∆t gives the precise value for the instantaneous velocity at t = 1. This is also the gradient of the tangent to the graph at t = 1. Instantaneous velocity is the instantaneous rate of change of position with respect to time. Although the scenario is unrealistic, these ideas show how you could answer the question of how fast you were going at t = 1. Given a function x(t) describing your position at time t, you could calculate your exact velocity at time t = 1. In fact, it is possible to calculate the instantaneous velocity at any value of t, obtaining a function which gives your instantaneous velocity at time t. This function is commonly denoted by x (t) or dx dt , and is known as the derivative of x(t) with respect to t. In this module, we will discuss derivatives. The more things change . . . Velocity is an important example of a derivative, but this is just one example. The world is full of quantities which change with respect to each other — and these rates of change can often be expressed as derivatives. It is often important to understand and predict how things will change, and so derivatives are important. Here are some examples of derivatives, illustrating the range of topics where derivatives are found: • Mechanics. We saw that the derivative of position with respect to time is velocity. Also, the derivative of velocity with respect to time is acceleration. And the derivative of momentum with respect to time is the (net) force acting on an object. • Civil engineering, topography. Let h(x) be the height of a road, or the altitude of a mountain, as you move along a horizontal distance x. The derivative h (x) with respect to distance is the gradient of the road or mountain. • Population growth. Suppose a population has size p(t) at time t. The derivative p (t) with respect to time is the population growth rate. The growth rates of human, animal and cell populations are important in demography, ecology and biology, respectively. • Economics. In macroeconomics, the rate of change of the gross domestic product (GDP) of an economy with respect to time is known as the economic growth rate. It is often used by economists and politicians as a measure of progress.
- 7. A guide for teachers – Years 11 and 12 • {7} • Mechanical engineering. Suppose that the total amount of energy produced by an engine is E(t) at time t. The derivative E (t) of energy with respect to time is the power of the engine. All of these examples arise from a more abstract question in mathematics: • Mathematics. Consider the graph of a function y = f (x), which is a curve in the plane. What is the gradient of a tangent to this graph at a point? Equivalently, what is the instantaneous rate of change of y with respect to x? In this module, we discuss purely mathematical questions about derivatives. In the three modules Applications of differentiation, Growth and decay and Motion in a straight line, we discuss some real-world examples. Therefore, although the Motivation section has focused on instantaneous velocity, which is an important motivating example, we now concentrate on calculating the gradient of a tangent to a curve. Content The gradient of secants and tangents to a graph Consider a function f : R → R and its graph y = f (x), which is a curve in the plane. We wish to find the gradient of this curve at a point. But first we need to define properly what we mean by the gradient of a curve at a point! The module Coordinate geometry defines the gradient of a line in the plane: Given a non-vertical line and two points on it, the gradient is defined as rise run . run rise Gradient of a line.
- 8. {8} • Introduction to differential calculus Now, given a curve defined by y = f (x), and a point p on the curve, consider another point q on the curve near p, and draw the line pq connecting p and q. This line is called a secant line. We write the coordinates of p as (x, y), and the coordinates of q as (x +∆x, y +∆y). Here ∆x represents a small change in x, and ∆y represents the corresponding small change in y. y x 0 x x + ∆x y = f(x) p = (x,y) ∆x q = (x + ∆x, y + ∆y) ∆y = f(x + ∆x) − f(x) Secant connecting points on the graph y = f (x) at x and x +∆x. As ∆x becomes smaller and smaller, the point q approaches p, and the secant line pq approaches a line called the tangent to the curve at p. We define the gradient of the curve at p to be the gradient of this tangent line. y x 0 x y = f(x) p = (x,y) Secants on y = f (x) approaching the tangent line at x. Note that, in this definition, the approximation of a tangent line by secant lines is just like the approximation of instantaneous velocity by average velocities, as discussed in the Motivation section. With this definition, we now consider how to compute the gradient of the curve y = f (x) at the point p = (x, y).
- 9. A guide for teachers – Years 11 and 12 • {9} Taking q = (x +∆x, y +∆y) as above, the secant line pq has gradient rise run = ∆y ∆x = f (x +∆x)− f (x) ∆x . Note that the symbol ∆ on its own has no meaning: ∆x and ∆y refer to change in x and y, respectively. You cannot cancel the ∆’s! As ∆x → 0, the gradient of the tangent line is given by lim ∆x→0 ∆y ∆x = lim ∆x→0 f (x +∆x)− f (x) ∆x . We also denote this limit by d y dx : lim ∆x→0 ∆y ∆x = d y dx . The notation d y dx indicates the instantaneous rate of change of y with respect to x, and is not a fraction. For our purposes, the expressions dx and d y have no meaning on their own, and the d’s do not cancel! The gradient of a secant is analogous to average velocity, and the gradient of a tangent is analogous to instantaneous velocity. Velocity is the instantaneous rate of change of position with respect to time, and the gradient of a tangent to the graph y = f (x) is the instantaneous rate of change of y with respect to x. Calculating the gradient of y = x2 Let us consider a specific function f (x) = x2 and its graph y = f (x), which is the standard parabola. To illustrate the ideas in the previous section, we will calculate the gradient of this curve at x = 1. We first construct secant lines between the points on the graph at x = 1 and x = 1+∆x, and calculate their gradients. y x 0 1 1.5 2 3 y = x2 ∆x = 2, gradient = 4 ∆x = 1, gradient = 3 ∆x = 0.5, gradient = 2.5 Gradients of secants from x = 1 to x = 1+∆x.
- 10. {10} • Introduction to differential calculus For instance, taking ∆x = 2, we consider the secant connecting the points at x = 1 and x = 3. Between these two points, f (x) increases from f (1) = 1 to f (3) = 9, giving ∆y = 8, and hence ∆y ∆x = 8 2 = 4. We compute gradients of secants for various values of ∆x in the following table. Secants of the parabola f (x) = x2 Secant between points ∆x ∆y = f (x +∆x)− f (x) Gradient of secant ∆y ∆x x = 1, x = 3 2 8 4 x = 1, x = 2 1 3 3 x = 1, x = 1.5 0.5 1.25 2.5 x = 1, x = 1.1 0.1 0.21 2.1 x = 1, x = 1.001 0.001 0.002001 2.001 As ∆x approaches 0, the gradients of the secants approach 2. It turns out that indeed the gradient of the tangent at x = 1 is 2. To see why, consider the interval of length ∆x, from x = 1 to x = 1+∆x. We have ∆y = f (1+∆x)− f (1) = (1+∆x)2 −12 = 2∆x +(∆x)2 , so that ∆y ∆x = 2∆x +(∆x)2 ∆x = 2+∆x. In the limit, as ∆x → 0, we obtain the instantaneous rate of change d y dx = lim ∆x→0 ∆y ∆x = lim ∆x→0 2+∆x = 2. (So, if you were riding your bike and your position was f (x) = x2 metres after x seconds, then your instantaneous velocity after 1 second would be 2 metres per second.)
- 11. A guide for teachers – Years 11 and 12 • {11} There’s nothing special about the point x = 1 or the function f (x) = x2 , as the following example illustrates. Example Let f (x) = x3 . What is the gradient of the tangent line to the graph y = f (x) at the point (2,8)? Solution The gradient at x = 2 is given by the limit lim ∆x→0 f (2+∆x)− f (2) ∆x = lim ∆x→0 (2+∆x)3 −23 ∆x = lim ∆x→0 8+12(∆x)+6(∆x)2 +(∆x)3 −8 ∆x = lim ∆x→0 12+6(∆x)+(∆x)2 = 12. Thus the gradient of the tangent line to y = f (x) at (2,8) is 12. Exercise 1 Using a similar method, find the gradient of the tangent line to y = x4 at (−1,1), and find the equation of this line. In the examples so far, we have been given a curve, and we have found the gradient of the curve at one particular point on the curve. But we can also find the gradient at all points simultaneously, as the next section illustrates. Definition of the derivative The method we used in the previous section to find the gradient of a tangent to a graph at a point can actually be used to work out the gradient everywhere, simultaneously. Example Let f (x) = x2 . What is the gradient of the tangent line to the graph y = f (x) at a general point (x, f (x)) on this graph?
- 12. {12} • Introduction to differential calculus Solution We calculate the same limit as in previous examples, using the variable x in place of a number: lim ∆x→0 f (x +∆x)− f (x) ∆x = lim ∆x→0 (x +∆x)2 − x2 ∆x = lim ∆x→0 x2 +2x(∆x)+(∆x)2 − x2 ∆x = lim ∆x→0 2x +∆x = 2x. Thus, the gradient of the tangent line at the point (x, f (x)) is 2x. The derivative of a function f (x) is the function f (x) which gives the gradient of the tangent to the graph y = f (x) at each value of x. It is often also denoted d y dx . Thus f (x) = d y dx = lim ∆x→0 f (x +∆x)− f (x) ∆x . The previous example shows that the derivative of f (x) = x2 is f (x) = 2x. Exercise 2 Show that the derivative of f (x) = x3 is f (x) = 3x2 . Functions which have derivatives are called differentiable. Not all functions are differ- entiable; in particular, to be differentiable, a function must be continuous. Almost all functions we meet in secondary school mathematics are differentiable. In particular, all polynomials, rational functions, exponentials, logarithms and trigonometric functions (such as sin, cos and tan) are differentiable. Derivatives of exponential, logarithmic and trigonometric functions are discussed in the two modules Exponential and logarithmic functions and The calculus of trigonometric functions. We shall say a little more about which functions are differentiable and which are not in the Appendix to this module. In the following example and exercises, we differentiate constant and linear functions.
- 13. A guide for teachers – Years 11 and 12 • {13} Exercise 3 Show that the derivative of a constant function f (x) = c, with c a real constant, is given by f (x) = 0. Example What is the derivative of the linear function f (x) = 3x +7? Solution We calculate f (x) = lim ∆x→0 f (x +∆x)− f (x) ∆x = lim ∆x→0 3(x +∆x)+7−3x −7 ∆x = lim ∆x→0 3∆x ∆x = 3. Thus f (x) = 3. (This answer makes sense, as the graph of y = f (x) is the line y = 3x +7, which has gradient 3.) Exercise 4 Show that the derivative of a linear function f (x) = ax + b, with a,b real constants, is f (x) = a. Notation for the derivative We have introduced two different notations for the derivative. Both are standard, and it is necessary to be proficient with both. The first notation is to write f (x) for the derivative of the function f (x). This functional notation was introduced by Lagrange, based on Isaac Newton’s ideas. The dash in f (x) denotes that f (x) is derived from f (x). The other notation is to write d y dx . This notation refers to the instantaneous rate of change of y with respect to x, and was introduced by Gottfried Wilhelm Leibniz, one of the dis- coverers of calculus. (The other discoverer was Isaac Newton. In fact, the question of who discovered calculus first was historically a point of great controversy. For more de- tails, see the History and applications section.)
- 14. {14} • Introduction to differential calculus If we have a function f (x) and its graph y = f (x), then the derivative f (x) is the gradient of the tangent of y = f (x), which is also the instantaneous rate of change d y dx . Thus the notations are equivalent: d y dx = lim ∆x→0 ∆y ∆x = lim ∆x→0 f (x +∆x)− f (x) ∆x = f (x). For example, we calculated earlier that the derivative of x2 was 2x. Thus y = x2 implies d y dx = 2x. Alternatively, f (x) = x2 implies f (x) = 2x. The two notations express the same result. Another common usage of Leibniz notation is to consider d dx as an operator, meaning ‘differentiate with respect to x’. So d y dx = d dx (y) means: take y, and differentiate with respect to x. An alternative way to denote differen- tiation of x2 using Leibniz notation would be d dx (x2 ) = 2x. At secondary school level, a ‘d’ or ‘dx’ or ‘d y’ has no meaning in itself; it only makes sense as part of a d dx or d y dx or similar.1 The d’s do not cancel. The two types of notation each have their advantages and disadvantages. It is also com- mon to mix notation, and we will do so in this module. For example, we can write d dx f (x) = f (x). Some derivatives So far in the examples and exercises we have found the following derivatives. f (x) f (x) constant c 0 linear ax +b a x2 2x x3 3x2 1 However, in more advanced pure mathematics, d is regarded as a more complicated object called a differ- ential operator, and objects like dx and d y are studied as examples of differential forms.
- 15. A guide for teachers – Years 11 and 12 • {15} Note that the example f (x) = ax + b includes the case f (x) = x, which has derivative f (x) = 1. Since we have seen that d dx (x) = 1, d dx (x2 ) = 2x and d dx (x3 ) = 3x2 , it is natural to conjecture that the derivative of xn is nxn−1 . We will now compute the derivative of f (x) = xn , for any positive integer n. To do so, we need the binomial expansion (a +b)n = an + n 1 an−1 b + n 2 an−2 b2 +···+ n n −2 a2 bn−2 + n n −1 abn−1 +bn . See the module The binomial theorem for details. We begin by using the binomial theorem to expand f (x +∆x) = (x +∆x)n = xn + n 1 xn−1 (∆x)+ n 2 xn−2 (∆x)2 +···+ n n −1 x(∆x)n−1 +(∆x)n . We can then compute f (x +∆x)− f (x) ∆x = (x +∆x)n − xn ∆x = n 1 xn−1 + n 2 xn−2 (∆x)+···+ n n −1 x(∆x)n−2 +(∆x)n−1 . The last line has a ∆x in every term except the first. Since n 1 = n, we have f (x) = lim ∆x→0 f (x +∆x)− f (∆x) ∆x = nxn−1 . The fact that the derivative of xn is nxn−1 holds more generally than just when n is a positive integer. The next exercise shows it is also true when n = −1. Exercise 14, later in this module, shows how to find the derivative of negative integer powers of x in general. Exercise 5 Prove that the derivative of f (x) = 1 x is f (x) = − 1 x2 . That is, prove that d dx (x−1 ) = −x−2 .
- 16. {16} • Introduction to differential calculus In fact, it’s also true that for any non-zero rational number (i.e., fraction) n, the derivative of f (x) = xn is f (x) = nxn−1 . See exercises 20 and 21 later in this module. Even more generally, for any real number a, including irrational a, the derivative of f (x) = xa is f (x) = axa−1 . It is not obvious how to even define what it means to raise a number to the power of an irrational number. For instance, 23 just means 2 × 2 × 2, and 2 7 5 just means 5 27, but what does 2 3 mean? In the module Exponential and logarithmic functions, we explore these issues, show how to define xa precisely for any real number a, and show that the derivative of xa is axa−1 . In summary, the following theorem is true. Theorem For any real number a, the derivative of f (x) = xa is f (x) = axa−1 , wherever f (x) is defined. Properties of the derivative We now consider various properties of differentiation. As we proceed, we will be able to differentiate wider and wider classes of functions. Throughout this section and the next, we will be manipulating limits as we compute derivatives. We therefore recall some basic rules for limits; see the module Limits and continuity for details. The following hold provided the limits lim x→a f (x) and lim x→a g(x) exist. • The limit of a sum (or difference) is the sum (or difference) of the limits: lim x→a f (x)± g(x) = lim x→a f (x)± lim x→a g(x). • The limit of a product is the product of the limits: lim x→a f (x)g(x) = lim x→a f (x) lim x→a g(x) . This includes the case of multiplication by a constant c: lim x→a c f (x) = c lim x→a f (x).
- 17. A guide for teachers – Years 11 and 12 • {17} The derivative of a constant multiple Suppose we want to differentiate 4x7 . Rather than returning to the definition of a deriva- tive, we can use the following theorem. Theorem Let f be a differentiable function and let c be a constant. Then the derivative of c f (x) is c f (x). That is, d dx c f (x) = c f (x). This fact can also be written in Leibniz notation as d dx c f = c d f dx . Proof The derivative of c f (x) is given by lim ∆x→0 c f (x +∆x)−c f (x) ∆x = c lim ∆x→0 f (x +∆x)− f (x) ∆x = c f (x). We may factor out the c, since it is just a constant. Example What is the derivative of 4x7 ? Solution The theorem tells us that the derivative of 4x7 is 4 times the derivative of x7 . Hence, d dx 4x7 = 4 d dx x7 = 4·7x6 = 28x6 . The derivative of a sum A function like f (x) = x3 + 5x2 can be differentiated from first principles; alternatively, we can use the following theorem.
- 18. {18} • Introduction to differential calculus Theorem Suppose both f and g are differentiable functions. Then the derivative of f (x)+ g(x) is f (x)+ g (x) and, similarly, the derivative of f (x)− g(x) is f (x)− g (x). That is, d dx f (x)+ g(x) = f (x)+ g (x), d dx f (x)− g(x) = f (x)− g (x). In Leibniz notation, these statements can be written as d dx f + g = d f dx + dg dx , d dx f − g = d f dx − dg dx . Proof We prove the first statement. The derivative of f (x)+ g(x) is given by lim ∆x→0 f (x +∆x)+ g(x +∆x) − f (x)+ g(x) ∆x = lim ∆x→0 f (x +∆x)− f (x) ∆x + lim ∆x→0 g(x +∆x)− g(x) ∆x = f (x)+ g (x). Note that, as both the limits in the second line exist, we are retrospectively justified in splitting the first limit into two pieces. Exercise 6 Adapt the above proof to prove the second statement of the theorem: the derivative of the difference of two functions is the difference of the derivatives. Example Find the derivative of f (x) = x3 +5x2 . Solution We have, using the above theorems, f (x) = d dx x3 +5x2 = d dx x3 + d dx 5x2 (derivative of a sum) = d dx x3 +5 d dx x2 (derivative of a constant multiple) = 3x2 +10x (derivative of xn ).
- 19. A guide for teachers – Years 11 and 12 • {19} Note. The solution to the previous example shows every individual step explicitly and states which theorems are used. Once proficient with these properties of the derivative, however, there is no need to justify each step in this way. It is common, for instance, to go straight from f (x) = x3 +5x2 to f (x) = 3x2 +10x. Exercise 7 Let f (x) = x5 + x x2 . Find f (x). Linearity of the derivative The two previous theorems (for the derivative of a sum and the derivative of a constant multiple) can be summarised as follows. If f ,g are differentiable functions and a,b are constants, then d dx a f (x)+b g(x) = a f (x)+b g (x). The same fact can be written in Leibniz notation as d dx a f +bg = a d f dx +b dg dx . This property is sometimes expressed by saying that ‘differentiation is linear’. The product, quotient and chain rules We now move to some more involved properties of differentiation. To summarise, so far we have found that: • the derivative of a constant multiple is the constant multiple of the derivative • the derivative of a sum is the sum of the derivatives • the derivative of a difference is the difference of the derivatives. However, it turns out that: • the derivative of a product f (x)g(x) is not the product of the derivatives • the derivative of a quotient f (x) g(x) is not the quotient of the derivatives • the derivative of the composition f (g(x)) is not the composition of the derivatives. The product, quotient and chain rules tell us how to differentiate in these three situa- tions. We consider the three rules in turn.
- 20. {20} • Introduction to differential calculus The product rule Theorem (Product rule) Let f ,g be differentiable functions. Then the derivative of their product is given by d dx f (x)g(x) = f (x)g (x)+ g(x) f (x). The product rule is also often written as d dx f g = f dg dx + g d f dx . Proof As before, we evaluate the limit which gives the derivative: d dx f (x)g(x) = lim ∆x→0 f (x +∆x)g(x +∆x)− f (x)g(x) ∆x . The trick is to add and subtract an extra term in the numerator, so that we can factorise and obtain some familiar-looking expressions: f (x +∆x)g(x +∆x)− f (x)g(x) = f (x +∆x)g(x +∆x)− f (x)g(x +∆x)+ f (x)g(x +∆x)− f (x)g(x) = f (x +∆x)− f (x) g(x +∆x)+ f (x) g(x +∆x)− g(x) . We can then rewrite the limit as lim ∆x→0 f (x +∆x)− f (x) ∆x g(x +∆x)+ f (x) g(x +∆x)− g(x) ∆x . Now, since the limit of a sum is the sum of the limits, and the limit of a product is the product of the limits, we obtain lim ∆x→0 f (x +∆x)− f (x) ∆x lim ∆x→0 g(x +∆x) + f (x) lim ∆x→0 g(x +∆x)− g(x) ∆x . We also used the fact that f (x) does not depend on ∆x. Recognising f (x) and g (x), and substituting ∆x = 0 into g(x +∆x), we obtain d dx f (x)g(x) = f (x)g(x)+ f (x)g (x), which is equivalent to the desired formula.
- 21. A guide for teachers – Years 11 and 12 • {21} Exercise 8 Starting from the fact that the derivative of x is 1, use the product rule to prove by induc- tion on n, that for all positive integers n, d dx xn = nxn−1 . Example Let f (x) = (x3 +2)(x2 +1). Find f (x). Solution We could expand out f (x) and differentiate term-by-term. Alternatively, with the prod- uct rule, we obtain f (x) = (x3 +2) d dx x2 +1 +(x2 +1) d dx x3 +2 = (x3 +2)·2x +(x2 +1)·3x2 = 5x4 +3x2 +4x. Exercise 9 Using the product rule, prove that in general, for a differentiable function f : R → R, the derivative of (f (x))2 with respect to x is 2 f (x) f (x). Exercise 10 By using the product rule, prove the following ‘extended product rule’: d dx f (x)g(x)h(x) = f (x)g(x)h(x)+ f (x)g (x)h(x)+ f (x)g(x)h (x). Generalise to the product of any number of functions. The chain rule The chain rule allows us to differentiate the composition of two functions. Recall from the module Functions II that the composition of two functions g and f is (f ◦ g)(x) = f (g(x)). We start with x, apply g, then apply f . The chain rule tells us how to differentiate such a function.
- 22. {22} • Introduction to differential calculus Theorem (Chain rule) Let f ,g be differentiable functions. Then the derivative of their composition is d dx f (g(x)) = f (g(x))g (x). In Leibniz notation, we may write u = g(x) and y = f (u) = f (g(x)); diagrammatically, x g → u f → y. Then the chain rule says that ‘differentials cancel’ in the sense that d y dx = d y du du dx . Proof To calculate the derivative, we must evaluate the limit d dx f (g(x)) = lim ∆x→0 f (g(x +∆x))− f (g(x)) ∆x . The trick is to multiply and divide by an extra term in the expression above, as shown, so that we obtain two expressions which both express rates of change: f (g(x +∆x))− f (g(x)) ∆x = f (g(x +∆x))− f (g(x)) g(x +∆x)− g(x) g(x +∆x)− g(x) ∆x . We can then rewrite the desired limit as lim ∆x→0 f (g(x +∆x))− f (g(x)) g(x +∆x)− g(x) lim ∆x→0 g(x +∆x)− g(x) ∆x . The ratio in the first limit expresses the change in the function f , from its value at g(x) to its value at g(x + ∆x), relative to the difference between g(x + ∆x) and g(x). So as ∆x → 0, this first term approaches the derivative of f at the point g(x), namely f (g(x)). The second limit is clearly g (x). We conclude that d dx f (g(x)) = f (g(x))g (x), as required. The proof above is not entirely rigorous: for instance, if there are values of ∆x close to zero such that g(x +∆x)− g(x) = 0, then we have division by zero in the first limit. How- ever, a fully rigorous proof is beyond the secondary school level. The next two examples illustrate ‘functional’ and ‘Leibniz’ methods of attacking the same problem using the chain rule.
- 23. A guide for teachers – Years 11 and 12 • {23} Example Let f (x) = (x7 − x2 )42 . Find f (x). Solution The function f (x) is the composition of the functions g(x) = x7 −x2 and h(x) = x42 , that is, f (x) = h(g(x)). We compute g (x) = 7x6 −2x, h (x) = 42x41 , and the chain rule gives f (x) = h (g(x))g (x) = 42(x7 − x2 )41 (7x6 −2x). Example Let y = (x7 − x2 )42 . Find d y dx . Solution Let u = x7 − x2 , so that y = u42 . We then have d y du = 42u41 , du dx = 7x6 −2x, and the chain rule gives d y dx = d y du du dx = 42u41 (7x6 −2x). Rewriting u in terms of x gives d y dx = 42(x7 − x2 )41 (7x6 −2x). Exercise 11 Find the derivative of (x2 +7)100 with respect to x. Exercise 12 In exercise 9, we proved that the derivative of (f (x))2 with respect to x is 2 f (x) f (x). Re-prove this fact using the chain rule.
- 24. {24} • Introduction to differential calculus Exercise 13 Prove the following ‘extended chain rule’: d dx f (g(h(x))) = f (g(h(x)))g (h(x))h (x). Generalise to the composition of any number of functions. The following exercise shows how, if you know the derivative of xn for a positive num- ber n, you can find the derivative of x−n . Exercise 14 Let g(x) = xn , where n is positive. Using the facts g (x) = nxn−1 and d dx 1 x = − 1 x2 and the chain rule, calculate d dx 1 xn . The quotient rule Theorem (Quotient rule) Let f ,g be differentiable functions. Then the derivative of their quotient is d dx f (x) g(x) = g(x) f (x)− f (x)g (x) (g(x))2 . Alternatively, we can write d dx f g = g d f dx − f dg dx g2 . Example Let f (x) = x2 +1 x2 −1 . What is f (x)? Solution Using the quotient rule, we have f (x) = (x2 −1) d dx (x2 +1)−(x2 +1) d dx (x2 −1) (x2 −1)2 = (x2 −1)·2x −(x2 +1)·2x (x2 −1)2 = −4x (x2 −1)2 .
- 25. A guide for teachers – Years 11 and 12 • {25} Example Let f (x) = x x2 +1 . Find f (x). Solution We first apply the quotient rule: f (x) = x2 +1 d dx (x)− x d dx x2 +1 x2 +1 . To differentiate x2 +1, we use the chain rule: d dx (x2 +1) 1 2 = 1 2 (x2 +1)− 1 2 d dx (x2 +1) = 1 2 (x2 +1)− 1 2 (2x) = x(x2 +1)− 1 2 . Now returning to f (x), we obtain f (x) = x2 +1·1− x · x(x2 +1)− 1 2 x2 +1 = (x2 +1)− x2 (x2 +1) 3 2 = (x2 +1)− 3 2 . The quotient rule can be proved using the product and chain rules, as the next two exer- cises show. Exercise 15 Let g be a differentiable function. Using the chain rule, show that d dx 1 g(x) = − g (x) (g(x))2 . (This is a generalisation of exercise 14.) Exercise 16 Using the previous exercise and the product rule, find the derivative of f (x) · 1 g(x) , and hence prove the quotient rule.
- 26. {26} • Introduction to differential calculus Summary of differentiation rules We can summarise the differentiation rules we have found as follows. They can be ex- pressed in both functional and Leibniz notation. First, the linearity of differentiation. Linearity of differentiation Functional notation Leibniz notation Constant multiple d dx c f (x) = c f (x) d dx c f = c d f dx Sum d dx f (x)+ g(x) = f (x)+ g (x) d dx f + g = d f dx + dg dx Difference d dx f (x)− g(x) = f (x)− g (x) d dx f − g = d f dx − dg dx We also have the product, quotient and chain rules. In Leibniz notation, these rules are often written with u,v rather than f ,g. Product, quotient and chain rules Functional notation Leibniz notation Product d dx f (x)g(x) = f (x)g (x)+ g(x) f (x) d dx uv = u dv dx + v du dx Quotient d dx f (x) g(x) = g(x) f (x)− f (x)g (x) (g(x))2 d dx u v = v du dx −u dv dx v2 Chain d dx f (g(x)) = f (g(x))g (x) d y dx = d y du du dx The tangent line to a graph Given a graph y = f (x), we have seen how to calculate the gradient of a tangent line to this graph. We can go further and find the equation of a tangent line. Consider the tangent line to the graph y = f (x) at x = a. This line has gradient f (a) and passes through the point (a, f (a)). Once we know a point on the line and its gradient, we can write down its equation: y − f (a) = f (a)(x − a). (See the module Coordinate geometry.)
- 27. A guide for teachers – Years 11 and 12 • {27} Example Find the equation of the tangent line to the graph y = 1 2 x2 at x = 3. Solution Letting f (x) = 1 2 x2 , we have f (x) = x, so f (3) = 9 2 and f (3) = 3. Thus the tangent line has gradient 3 and passes through (3, 9 2 ), and is given by y − 9 2 = 3(x −3) or, equivalently, y = 3x − 9 2 . Exercise 17 What is the equation of the tangent line to the graph of y = 9− x2 at x = 3 2 2 ? The second derivative Given a function f (x), we can differentiate it to obtain f (x). It can be useful for many purposes to differentiate again and consider the second derivative of a function. In functional notation, the second derivative is denoted by f (x). In Leibniz notation, letting y = f (x), the second derivative is denoted by d2 y dx2 . The placement of the 2’s in the notation d2 y dx2 may appear unusual. We consider that we have applied the differentiation operator d dx twice to y: d dx 2 y = d2 dx2 y = d2 y dx2 ; or that we have applied the differentiation operator d dx to d y dx : d2 y dx2 = d dx d y dx . As we will see in the module Applications of differentiation, the second derivative can be very useful in curve-sketching. The second derivative determines the convexity of the graph y = f (x) and can, for example, be used to distinguish maxima from minima. The second derivative can also have a physical meaning. For example, if x(t) gives posi- tion at time t, then x (t) is the velocity and the second derivative x (t) is the acceleration at time t. This is discussed in the module Motion in a straight line.
- 28. {28} • Introduction to differential calculus Example Find the second derivative of f (x) = x2 . Solution We have f (x) = 2x, and so f (x) = 2. This example implies that, if you ride your bike and your position is x2 metres after x sec- onds, then your acceleration is a constant 2 m/s2 . Example Let y = x7 +3x5 + x 3 2 . Find d2 y dx2 . Solution The first derivative is d y dx = 7x6 +15x4 + 3 2 x 1 2 , so the second derivative is d2 y dx2 = 42x5 +60x3 + 3 4 x− 1 2 . Exercise 18 Let f (x) = (x2 +7)100 , as in exercise 11. What is f (x)? Exercise 19 Suppose the position of an object at time t is given by x(t) = 1−7t +(t −5)4 . Show that x (t) ≥ 0 for all t, so that acceleration is always non-negative. Differentiation of inverses A clever use of the chain rule arises when we have a function f and its inverse func- tion f −1 . Refer to the module Functions II for a discussion of inverse functions.
- 29. A guide for teachers – Years 11 and 12 • {29} Letting y = f (x), we can express x as the inverse function of y: y = f (x), x = f −1 (y). The composition of f and its inverse f −1 , by definition, is just x. That is, f −1 (f (x)) = x. We can think of this diagrammatically as x f → y f −1 → x. Using the chain rule, we can differentiate this composition of functions to obtain dx dx = dx d y d y dx . The derivative dx dx of x with respect to x is just 1, so we obtain the important formula 1 = dx d y d y dx , which can also be expressed as d y dx = 1 dx d y or dx d y = 1 d y dx . This formula allows us to differentiate inverse functions, as in the following example. Example Let y = 3 x. Find d y dx . Solution (We assume that we only know the derivative of xn when n is a positive integer.) The inverse function of the cube-root function is the cube function: if y = 3 x, then x = y3 . We know the derivative of the cube function, dx d y = 3y2 . We use this to find d y dx : d y dx = 1 dx d y = 1 3y2 . Substituting y = 3 x = x 1 3 gives d y dx = 1 3x 2 3 = 1 3 x− 2 3 , as expected. The following exercises generalise the above example to find the derivative of n x = x 1 n , and then of any rational power of x.
- 30. {30} • Introduction to differential calculus Exercise 20 Let n be a positive integer. Using the fact that the derivative of xn is nxn−1 , prove that the derivative of x 1 n is 1 n x 1 n −1 . Exercise 21 Using the chain rule and the previous exercise, prove that for any rational number p q the derivative of x p q is p q x p q −1 . (Here p,q are integers and q > 0.) Implicit differentiation Implicit differentiation is a powerful technique to find an instantaneous rate of change d y dx when there is an equation relating x and y. It applies even when y is not a function of x. All that is required is that there is an equation relating x and y. For example, consider the curve in the plane described by the equation x2 + y2 = 9. This is a circle centred at the origin, of radius 3. x y 0 3 3 –3 –3 (x, y) The circle x2 + y2 = 9. Note that y is not a function of x! The circle fails the vertical-line test; we have a relation, not a function. (See the module Functions I for a discussion of functions and relations.) For a given value of x, there may be two distinct values of y: x2 + y2 = 9 ⇐⇒ y = ± 9− x2. Although y is not a function of x, we can say that the equation x2 + y2 = 9 expresses y as an implicit function of x. Suppose you want to find the gradient of the tangent to this circle at a point (x, y). One approach would be to consider either f1(x) = 9− x2 or f2(x) = − 9− x2, depending on whether y is positive or negative, and then differentiate. (We saw the derivative of 9− x2 in exercise 17.)
- 31. A guide for teachers – Years 11 and 12 • {31} The much better approach of implicit differentiation is to differentiate both sides of the equation x2 + y2 = 9 with respect to x. Differentiating y2 requires the chain rule, since y2 is a function of y and y is a function of x: d dx (y2 ) = d d y (y2 ) d y dx = 2y d y dx . We can now differentiate each term of x2 + y2 = 9, and we obtain 2x +2y d y dx = 0. Note the power of implicit differentiation: this equation involving d y dx is valid for all points on the circle except y = 0. When y = 0, we may solve for d y dx and obtain d y dx = − x y . Without implicit differentiation, we would not have found such a neat formula! From this formula, we can see that if x, y have the same sign, then the gradient of the tangent is negative; while if x, y are of opposite sign, then the gradient is positive. Exercise 22 Give a geometric argument why the tangent at the point (x, y) to a circle centred at the origin has gradient − x y . Example What is the gradient of the curve y2 = x3 −7x at the points where x = 4? Solution When x = 4, we have y2 = 64−28 = 36, so y = ±6. Implicit differentiation of y2 = x3 −7x gives 2y d y dx = 3x2 −7. So when y = 0, we obtain d y dx = 3x2 −7 2y . Hence: • At (x, y) = (4,6), we have d y dx = 3·42 −7 2·6 = 41 12 . • At (x, y) = (4,−6), we have d y dx = 3·42 −7 2·(−6) = − 41 12 . The curve in the previous example is called an elliptic curve.
- 32. {32} • Introduction to differential calculus Exercise 23 Consider the hyperbola x2 − y2 = 5. a By writing y in terms of x, find the gradient of the tangent to the hyperbola at (3,−2). b Find the same gradient by implicit differentiation. Links forward Higher derivatives So far in this module, we have looked at first and second derivatives. But there’s no need to stop differentiating after doing it twice! Given a function f (x), if it can be differentiated k times, then we write the kth derivative as f (k) (x). In Leibniz notation, the kth derivative of y with respect to x is denoted by dk y dxk . Higher derivatives are useful for finding polynomial approximations to functions, as we see in the next section. Exercise 24 Let n be a positive integer. What is the nth derivative of the function f (x) = xn ? Approximations of functions We have seen that the equation of the tangent line to the graph y = f (x) at the point x = a is given by y = f (a)+ f (a)(x − a). Therefore, if we define a linear function T1(x) as T1(x) = f (a)+ f (a)(x − a), then T1(x) approximates f (x) near x = a. In fact, since T1(x) = f (a), we have T1(a) = f (a) and T1(a) = f (a). So the functions T1(x) and f (x) agree at x = a, and their derivatives also agree at x = a. In this sense, T1(x) is the ‘best linear approximation’ to f (x) at x = a.
- 33. A guide for teachers – Years 11 and 12 • {33} However, there’s no reason to stop at linear approximations: we might ask for the best quadratic approximation to f (x) at x = a. Consider the function T2(x) = f (a)+ f (a)(x − a)+ f (a) 2 (x − a)2 . This is a quadratic function. Its derivatives are T2(x) = f (a)+ f (a)(x − a) and T2 (x) = f (a), so T2(a) = f (a), T2(a) = f (a) and T2 (a) = f (a). That is, T2(x) and f (x) agree at x = a, as do their first and second derivatives. In this sense, T2(x) is the best quadratic approximation to f (x) at x = a. In general, one can check that the function Tk(x) = f (a)+ f (a)(x − a)+ f (a) 2! (x − a)2 + f (3) (a) 3! (x − a)3 +···+ f (k) (a) k! (x − a)k is a polynomial of degree k which agrees with f (x) at x = a, along with all its first k derivatives. That is, Tk(a) = f (a), Tk(a) = f (a), Tk (a) = f (a), ..., T (k) (a) = f (k) (a). In this sense, Tk(x) is the best polynomial approximation of degree k to f (x) at x = a. The polynomials Tk(x), called Taylor polynomials, are very important throughout math- ematics and science for approximating functions. Taylor polynomials and their general- isations are studied extensively in university-level mathematics courses. The following diagram shows some of the first few Taylor polynomials approximating the function f (x) = sinx near x = 0. Note how they provide successively better approx- imations to f (x). (We will see how to differentiate sinx in the module The calculus of trigonometric functions.) x y 0 1 2𝜋𝜋 –1 –2𝜋 –𝜋 y = T1(x) y = T5(x) y = sin x y = T3(x) Taylor polynomials approximating sinx at x = 0.
- 34. {34} • Introduction to differential calculus History and applications The discoverers of calculus Today it is generally believed that calculus was discovered independently in the late 17th century by two great mathematicians: Isaac Newton and Gottfried Leibniz. However, the dispute over who first discovered calculus became a major scandal around the turn of the 18th century. Like most scientific discoveries, the discovery of calculus did not arise out of a vacuum. In fact, many mathematicians and philosophers going back to ancient times made dis- coveries relating to calculus. The ancient Greeks made many discoveries that we would today think of as part of calcu- lus — however, mostly integral calculus, which will be discussed in the module Integra- tion. Indian mathematicians in Kerala had developed Taylor polynomials for functions like sinx and cosx before 1500. (See the article Was calculus invented in India? listed in the References section.) In the early 17th century, Fermat developed a method called adequality for finding where the derivative of a function is zero, that is, for solving f (x) = 0. But it was not until Newton and Leibniz that gradients of tangents to curves could be calculated in general. The Newton–Leibniz controversy Newton described his version of differential calculus as ‘the method of fluxions’. He wrote a paper on fluxions in 1666, but like many of his works, it was not published until decades later. His magnum opus Philosophiae naturalis principia mathematica (Math- ematical principles of natural philosophy) was published in 1687. This work includes his theories of motion and gravitation, but does not include much calculus explicitly — although there is some explanation of calculus at the beginning, and Newton certainly used calculus to formulate his theories. Nonetheless, Newton’s ‘method of fluxions’ did not explicitly appear in print until 1693. Leibniz, on the other hand, published his first paper on calculus in 1684 — and claimed to have discovered calculus in the 1670s. From the published record, at least, Leibniz seemed to have discovered calculus first. While Newton and Leibniz initially had a cordial relationship, Leibniz and his followers did not take kindly to a statement made by the English mathematician John Wallis. With a rather xenophobic and quarrelsome character, Wallis fought priority disputes on behalf
- 36. {36} • Introduction to differential calculus The most scathing criticism perhaps came from Bishop Berkeley (The Analyst, 1734), who ridiculed fluxions and infinitesimals: And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quan- tities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities? It was not until over a century later that ideas like limits were formally introduced, and put on a firm mathematical footing, so that today we present the derivative as f (x) = lim ∆x→0 f (x +∆x)− f (x) ∆x . As with many branches of mathematics, the way that calculus is taught and learned bears little relation to its historical development. Appendix Functions differentiable and not We mentioned earlier that not all functions have derivatives. We now say a little more, although a full discussion of these issues is the subject matter of higher level university courses on analysis. For a function to have a derivative, it must first of all be continuous — you must be able to draw the graph without taking your pen off the page! More rigorously, continuity at a point x = a means that we must have lim x→a f (x) = f (a). Roughly speaking, a function will only have a derivative where its graph is smooth, and can be drawn on the page without sharp corners. For instance, consider the absolute value function f (x) = |x| = x if x ≥ 0, −x if x < 0. The following diagram shows the graph of y = f (x).
- 37. A guide for teachers – Years 11 and 12 • {37} x y 0 y = f(x) Graph of f (x) = |x|. For x > 0, we have f (x) = x and so f (x) = 1. For x < 0, we have f (x) = −x and so f (x) = −1. However, at x = 0 we have a problem. x y 0 1 –1 y = f’(x) Graph of f (x). By definition, if f (0) were to exist, f (0) = lim ∆x→0 f (0+∆x)− f (0) ∆x = lim ∆x→0 |∆x| ∆x . Unfortunately, for any ∆x > 0 we have |∆x| ∆x = 1, while for any ∆x < 0 we have |∆x| ∆x = −1. So if ∆x approaches 0 from above, the limit is 1, but if ∆x approaches 0 from below, the limit is −1. Therefore, the limit does not exist, and f (0) is not defined. The function f (x) = |x| is therefore differentiable at all x = 0, but is not differentiable at x = 0. It is differentiable almost everywhere, but not everywhere. The absolute value function, however, is rather tame in comparison to examples like the following. Consider the infinite series f (x) = cos(πx)+ 1 2 cos(3πx)+ 1 22 cos(32 πx)+··· = ∞ n=0 1 2n cos(3n πx). The following graphs show a few partial sums of this series.
- 38. {38} • Introduction to differential calculus y cos Πx x y y cos Πx 1 2 cos 3Πx x y y cos Πx 1 2 cos 3Πx 1 4 cos 9Πx x y y cos Πx 1 2 cos 3Πx ... 1 210 cos 310 Πx x y Graphs of approximations to a nowhere-differentiable function. Note how the functions oscillate more rapidly and the graphs become ‘bumpier’ as each term is added. In the limit, the function oscillates so wildly that, although it remains continuous, it is too bumpy for the derivative to exist. It can be shown that this function is continuous everywhere, but is differentiable nowhere!
- 39. A guide for teachers – Years 11 and 12 • {39} Answers to exercises Exercise 1 Let f (x) = x4 . We first compute f (−1+∆x) = (−1+∆x)4 = 1−4(∆x)+6(∆x)2 −4(∆x)3 +(∆x)4 and f (−1) = 1. The gradient at x = −1 is then given by lim ∆x→0 f (−1+∆x)− f (−1) ∆x = lim ∆x→0 −4(∆x)+6(∆x)2 −4(∆x)3 +(∆x)4 ∆x = lim ∆x→0 −4+6(∆x)−4(∆x)2 +(∆x)3 = −4. The tangent line at (−1,1) has gradient −4, and hence has equation y −1 = −4(x +1) or, equivalently, y = −4x −3. Exercise 2 Let f (x) = x3 . We compute f (x) = lim ∆x→0 f (x +∆x)− f (x) ∆x = lim ∆x→0 (x +∆x)3 − x3 ∆x = lim ∆x→0 3x2 (∆x)+3x(∆x)2 +(∆x)3 ∆x = lim ∆x→0 3x2 +3x(∆x)+(∆x)2 = 3x2 . Exercise 3 Let f (x) = c. Then, for any x and ∆x, we have f (x + ∆x) − f (x) = c − c = 0. Hence, the derivative is f (x) = lim ∆x→0 f (x +∆x)− f (x) ∆x = lim ∆x→0 0 = 0. Exercise 4 Let f (x) = ax +b. Then f (x) = lim ∆x→0 f (x +∆x)− f (x) ∆x = lim ∆x→0 a(x +∆x)+b − ax −b ∆x = lim ∆x→0 a(∆x) ∆x = lim ∆x→0 a = a.
- 40. {40} • Introduction to differential calculus Exercise 5 Let f (x) = 1 x . We first compute the quotient f (x +∆x)− f (x) ∆x = 1 ∆x 1 x +∆x − 1 x = 1 ∆x · x −(x +∆x) x(x +∆x) = −∆x (∆x)x(x +∆x) = − 1 x(x +∆x) . Hence, the derivative is f (x) = lim ∆x→0 f (x +∆x)− f (x) ∆x = lim ∆x→0 − 1 x(x +∆x) = − 1 x2 . Thus the derivative of f (x) = 1 x is f (x) = − 1 x2 . Exercise 6 The derivative of f (x)− g(x) is given by lim ∆x→0 f (x +∆x)− g(x +∆x) − f (x)− g(x) ∆x = lim ∆x→0 f (x +∆x)− f (x) ∆x − lim ∆x→0 g(x +∆x)− g(x) ∆x = f (x)− g (x). Hence, the derivative of f (x)− g(x) is f (x)− g (x). Exercise 7 Rewriting f (x) as x3 + x− 3 2 , we obtain f (x) = 3x2 − 3 2 x− 5 2 . Exercise 8 We assume that the derivative of x is 1. We now prove that, if the derivative of xn is nxn−1 , then the derivative of xn+1 is (n +1)xn . To do this we use the product rule: d dx xn+1 = d dx xn · x = xn d dx x + x d dx xn = xn ·1+ x ·nxn−1 = (n +1)xn . It follows by induction that d dx (xn ) = nxn−1 , for all positive integers n.
- 41. A guide for teachers – Years 11 and 12 • {41} Exercise 9 Using the product rule, we have d dx f (x)2 = d dx f (x)· f (x) = f (x)· d dx f (x) + f (x)· d dx f (x) = f (x) f (x)+ f (x) f (x) = 2 f (x) f (x). Exercise 10 We first use the product rule on the product of f (x)g(x) and h(x): d dx f (x)g(x)h(x) = d dx f (x)g(x) h(x)+ f (x)g(x) d dx h(x) . Then we use the product rule on f (x)g(x): d dx f (x)g(x)h(x) = f (x)g(x)+ f (x)g (x) h(x)+ f (x)g(x)h (x) = f (x)g(x)h(x)+ f (x)g (x)h(x)+ f (x)g(x)h (x). For a general product f1(x) f2(x)··· fn(x), the derivative is a sum of n terms, with fi (x) occurring in the ith term: d dx f1(x) f2(x)··· fn(x) = f1(x) f2(x)··· fn(x)+ f1(x) f2(x) f3(x)··· fn(x)+···+ f1(x)··· fn−1(x) fn(x). Exercise 11 Let f (x) = (x2 +7)100 = g(h(x)), where g(x) = x100 and h(x) = x2 +7. Then g (x) = 100x99 and h (x) = 2x, so by the chain rule f (x) = g (h(x))h (x) = 100(x2 +7)99 ·2x = 200x(x2 +7)99 . Exercise 12 We can write f (x)2 as h(f (x)) where h(x) = x2 . The chain rule then gives d dx h(f (x)) = h (f (x)) f (x) = 2 f (x) f (x). Exercise 13 We first think of f (g(h(x))) as the composition of f (x) and g(h(x)), so the chain rule gives d dx f (g(h(x))) = f (g(h(x))) d dx g(h(x)).
- 42. {42} • Introduction to differential calculus Then using the chain rule again gives d dx f (g(h(x))) = f (g(h(x)))g (h(x))h (x). In general, for the composition of n functions f1 ◦ f2 ◦···◦ fn, the derivative is a product of n factors, and the ith factor is fi (fi+1(···(fn(x))···)). Exercise 14 Let g(x) = xn and h(x) = 1 x , so that 1 xn = h(g(x)). Then g (x) = nxn−1 and h (x) = − 1 x2 . By the chain rule, d dx 1 xn = d dx h(g(x)) = h (g(x))g (x) = − 1 (g(x))2 g (x) = − 1 x2n nxn−1 = −nx−n−1 . Thus, the derivative of x−n is −nx−n−1 . Exercise 15 Let h(x) = 1 x . Then h (x) = − 1 x2 and 1 g(x) = h(g(x)). By the chain rule, d dx 1 g(x) = d dx h(g(x)) = h (g(x))g (x) = − 1 (g(x))2 g (x) = − g (x) (g(x))2 . Exercise 16 Since the derivative of 1 g(x) is − g (x) (g(x))2 , the product rule gives d dx f (x) g(x) = 1 g(x) d dx f (x) + f (x) d dx 1 g(x) = f (x) g(x) + f (x) − g (x) (g(x))2 = g(x) f (x)− f (x)g (x) (g(x))2 . This is the quotient rule. Exercise 17 Let y = 9− x2. We differentiate: d y dx = 1 2 (9− x2 )− 1 2 ·(−2x) = −x 9− x2 . So, at x = 3 2 2 , we have y = 3 2 2 and d y dx = −3 2 2 2 3 2 = −1. The tangent line has gradient −1 and passes through the point (3 2 2 , 3 2 2 ), and hence has equation y = −x +3 2.
- 43. A guide for teachers – Years 11 and 12 • {43} Exercise 18 From f (x) = (x2 + 7)100 , we have f (x) = 200x(x2 + 7)99 . Using the product and chain rules, we obtain f (x) = 200x d dx (x2 +7)99 +(x2 +7)99 d dx 200x = 200x ·99(x2 +7)98 ·2x +(x2 +7)99 ·200 = 200(x2 +7)98 (199x2 +7). Exercise 19 We compute the derivatives of x(t) = 1−7t +(t −5)4 with respect to t: x (t) = −7+4(t −5)3 x (t) = 12(t −5)2 . Since squares are non-negative, we have x (t) ≥ 0 for all t. That is, the acceleration is always non-negative. Exercise 20 Let y = x 1 n , where n is a positive integer. We wish to find d y dx . We have x = yn , and so dx d y = nyn−1 . Thus d y dx = 1 dx d y = 1 nyn−1 = 1 n y1−n , and substituting y = x 1 n gives d y dx = 1 n x 1−n n = 1 n x 1 n −1 , as expected. Exercise 21 Let y = x p q , where p,q are integers with q > 0. We wish to find d y dx . We let u = x 1 q . Then y = up and, by the previous exercise, du dx = 1 q x 1 q −1 . The chain rule then gives d y dx = d y du du dx = pup−1 · 1 q x 1 q −1 = p q up−1 x 1 q −1 . Substituting u = x 1 q gives d y dx = p q x p−1 q x 1 q −1 = p q x p q −1 .
- 44. {44} • Introduction to differential calculus Exercise 22 If the circle is centred at the origin, then the radius of the circle from (0,0) to (x, y) has gradient y x . The tangent to the circle is perpendicular to the radius, and hence its gradient is the negative reciprocal of y x , that is, the gradient is −x y . Exercise 23 a From y2 = x2 −5, we have y = ± x2 −5. As we want to include the point (3,−2), we take the negative square root and consider y = − x2 −5. Then d y dx = − 1 2 (x2 −5)− 1 2 ·2x = −x x2 −5 . At x = 3, we have d y dx = −3 4 = −3 2 . b Implicit differentiation of x2 − y2 = 5 gives 2x −2y d y dx = 0, and so d y dx = x y . Hence, at the point (3,−2), we have d y dx = −3 2 . Exercise 24 The kth derivative of xn is n(n −1)···(n −k +1)xn−k , and hence the nth derivative is the constant n!. References • David Bressoud, ‘Was calculus invented in India?’, College Mathematics Journal 33 (2002), 2–13. • Alfred Rupert Hall, Philosophers at War: The Quarrel Between Newton and Leibniz, Cambridge University Press, 1980.