Elementary Real Analysis, 2Nd Edition (2008) Section 7.2
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ClassicalRealAnalysis.com Chapter 7 DIFFERENTIATION 7.1 Introduction Calculus courses succeed in conveying an idea of what a derivative is, and the students develop many technical skills in computations of derivatives or applications of them. We shall return to the subject of derivatives but with a different objective. Now we wish to see a little deeper and to understand the basis on which that theory develops. Much of this chapter will appear to be a review of the subject of derivatives with more attention paid to the details now and less to the applications. Some of the more advanced material will be, however, completely new. We start at the beginning, at the rudiments of the theory of derivatives. 7.2 The Derivative Let f be a function defined on an interval I and let x0 and x be points of I. Consider the difference quotient determined by the points x0 and x: f(x) f(x ) − 0 , (1) x x − 0 representing the average rate of change of f on the interval with endpoints at x and x0. 396 Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) Section 7.2. The Derivative ClassicalRealAnalysis.com 397 f f(x) f(x0) x0 x Figure 7.1. The chord determined by (x, f(x)) and (x0, f(x0)). In Figure 7.1 this difference quotient represents the slope of the chord (or secant line) determined by the points (x, f(x)) and (x0, f(x0)). This same picture allows a physical interpretation. If f(x) represents the distance a point moving on a straight line has moved from some fixed point in time x, then f(x) f(x ) − 0 represents the (net) distance it has moved in the time interval [x0, x], and the difference quotient (1) represents the average velocity in that time interval. Suppose now that we fix x0, and allow x to approach x0. We learn in elementary calculus that if f(x) f(x ) lim − 0 x x0 x x → − 0 exists, then the limit represents the slope of the tangent line to the graph of the function f at the point (x0, f(x0)). In the setting of motion, the limit represents instantaneous velocity at time x0. The derivative owes its origins to these two interpretations in geometry and in the physics of motion, but now completely transcends them; the derivative finds applications in nearly every part of mathematics and the sciences. We shall study the structure of derivatives, but with less concern for computations and applications than we would have seen in our calculus courses. Now we wish to understand the notion and see why it has the properties used in the many computations and applications of the calculus. Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) 398 ClassicalRealAnalysis.com Differentiation Chapter 7 7.2.1 Definition of the Derivative We begin with a familiar definition. Definition 7.1: Let f be defined on an interval I and let x I. The derivative of f at x , denoted by 0 ∈ 0 f ′(x0), is defined as f(x) f(x0) f ′(x0) = lim − , (2) x x0 x x → − 0 provided either that this limit exists or is infinite. If f ′(x0) is finite we say that f is differentiable at x0. If f is differentiable at every point of a set E I, we say that f is differentiable on E. When E is all of I, we simply say that f is a differentiable function.⊂ Note. We have allowed infinite derivatives and they do play a role in many studies, but differentiable always refers to a finite derivative. Normally the phrase “a derivative exists” also means that that derivative is finite. Example 7.2: Let f(x) = x2 on R and let x R. If x R, x = x , then 0 ∈ ∈ 6 0 f(x) f(x ) x2 x2 (x x )(x + x ) − 0 = − 0 = − 0 0 . x x x x (x x ) − 0 − 0 − 0 Since x = x , the last expression equals x + x , so 6 0 0 f(x) f(x0) lim − = lim (x + x0) = 2x0, x x0 x x x x0 → − 0 → 2 establishing the formula, f ′(x0) = 2x0 for the function f(x) = x . ◭ Let us take a moment to clarify the definition when the interval I contains one or both of its endpoints. Suppose I = [a, b]. For x0 = a (or x0 = b), the limit in (2) is just a one-sided, or unilateral, limit. The function f is defined only on [a, b] so we cannot consider points outside of that interval. Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) Section 7.2. The Derivative ClassicalRealAnalysis.com 399 This brings us to another point. It can happen that a function that is not differentiable at a point x0 does satisfy the requirement of (2) from one side of x . This means that the limit in (2) exists as x x 0 → 0 from that side. We present a formal definition. Definition 7.3: Let f be defined on an interval I and let x I. The right-hand derivative of f at x , 0 ∈ 0 denoted by f+′ (x0) is the limit f(x) f(x0) f+′ (x0) = lim − , x x0+ x x → − 0 provided that one-sided limit exists or is infinite. Similarly, the left-hand derivative of f at x0, f ′ (x0), is the limit − f(x) f(x0) f ′ (x0) = lim − . − x x0 x x → − − 0 Observe that, if x0 is an interior point of I, then f ′(x0) exists if and only if f+′ (x0) = f ′ (x0). (See Exercise 7.2.8) − Example 7.4: Let f(x) = x on R. Let us consider the differentiability of f at x0 = 0. The difference quotient (1) becomes | | f(x) f(0) x 1, if x > 0 − = | | = x 0 x 1, if x < 0. − − Thus x f+′ (0) = lim | | = 1 x x0+ x → while x f ′ (0) = lim | | = 1. x x0 x − → − − The function has different right-hand and left-hand derivatives at x0 = 0 so is not differentiable at x0 = 0. ◭ Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) 400 ClassicalRealAnalysis.com Differentiation Chapter 7 Figure 7.2. A function trapped between x2 and x2. − Example 7.5: (A “trapping principle”) Let f be any function defined in a neighborhood I of zero. Suppose f satisfies the inequality f(x) x2 for all x I. Thus, the graph of f is “trapped” between the parabolas y = x2 and y = x2. In| particular,| ≤ ∈ − f(0) = 0. The difference quotient computed for x0 = 0 becomes f(x) f(0) f(x) − = , x 0 x − from which we calculate f(x) x2 = x x ≤ x | | so f(x ) lim lim x = 0. x 0 x ≤ x 0 | | → → Thus f(x) lim = 0. x 0 x → As a result, f ′(0) = 0. Figure 7.2 illustrates the principle. ◭ Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) Section 7.2. The Derivative ClassicalRealAnalysis.com 401 Higher-Order Derivatives When a function f is differentiable on I, it is possible that its derivative f ′ is also differentiable. When this is the case, the function f ′′ = (f ′)′ is called the second derivative of the function (n+1) (n) (n) f. Inductively, we can define derivatives of all orders: f = (f )′ (provided f is differentiable). (2) (3) When n is small, it is customary to use the convenient notation f ′′ for f , f ′′′ for f etc. Notation It is useful to have other notations for the derivative of a function f. Common notations are df dy dx and dx (when the function is expressed in the form y = f(x)). Another notation that is useful is Df. These alternate notations along with slight variations are useful for various calculations. You are no doubt familiar with such uses—the convenience of writing dy dy du = dx du dx when using the chain rule, or viewing D as an operator in solving linear differential equations. Notation can be important at times. Consider, for example, how difficult it would be to perform a simple arithmetic calculation such as the multiplication (104)(90) using Roman numerals (CIV)(XC)! Exercises 7.2.1 You might be familiar with a slightly different formulation of the definition of derivative. If x0 is interior to I, then for h sufficiently small, the point x0 + h is also in I. Show that expression (2) then reduces to ′ f(x0 + h) f(x0) f (x0) = lim − . h→0 h Repeat Examples 7.2 and 7.4 using this formulation of the derivative. See Note 161 7.2.2 Let c R. Calculate the derivatives of the functions g(x) = c and k(x) = x directly from the definition of derivative.∈ 7.2.3 Check the differentiability of each of the functions below at x0 = 0. (a) f(x) = x x | | Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) 402 ClassicalRealAnalysis.com Differentiation Chapter 7 (b) f(x) = x sin x−1 (f(0) = 0) (c) f(x) = x2 sin x−1 (f(0) = 0) x2, if x rational (d) f(x) = 0, if x irrational x2, if x 0 7.2.4 Let f(x) = ax, if x≥ < 0 (a) For which values of a is f differentiable at x = 0? (b) For which values of a is f continuous at x = 0? (c) When f is differentiable at x = 0, does f ′′(0) exist? 7.2.5 For what positive values of p is the function f(x) = x p differentiable at 0? | | 7.2.6 A function f has a symmetric derivative at a point if ′ f(x + h) f(x h) fs(x) = lim − − h→0 2h ′ ′ ′ exists.