3
Differentiation Chapter Rules
In this chapter we learn the main calculational tools that allow us to nd derivatives faster and more easily than by working directly with limits. These will be used in the applications seen for the rest of the semester, starting in this chapter in Sections 3, 10 and 11. By the end of this chapter, you should be able to answer all the Concept Check Questions except 2(k-t), and all of the True-False Quiz Questions at the end of the chapter on pages 261. Note that we do not cover Section 8.
3.1
Derivatives of Polynomial and Exponential Functions
Our collection of efciently methods for computing derivatives starts with polynomials and exponential functions. Much as with limits, we do this by rst dealing with a few simple functions, and then using rules for handling constant multiples, sums and differences. However, derivatives of products and quotients do not follow quite the same pattern as with limits, so we leave them until the next section. Since the slope of a straight line y = mx + c is the constant m, it is easy to check that the derivative of f (x) = mx + c is m, for any constants m and c. It is often convenient to use notation directly with formulas, without naming the functions, so to illustrate several notations: Theorem (Derivatives of Linear Functions). The derivative of the linear function f (x) = mx + c is f (x) = (mx + c) = d (mx + c) = m. dx
The two most basic special cases are when the function is a constant c or just x: (c) = d (c) = 0, dx (x) = d (x) = 1. dx
Theorem (The Power Rule). For any positive integer n, (xn ) = dn (x ) = n xn1 dx 3.1.1
CHAPTER 3. DIFFERENTIATION RULES
3.1.2
This includes f (x) = x1 = x and f (x) = x0 = 1, cases seen above. We have also almost seen this for f (x) = x2 in examples above with quadratics. Rather than do that example, let us look at n = 3, which hints at how to do this calculation for any n. Example (A). Calculate the derivative of f (x) = x3 . It is convenient in this case to use the rst formula for the derivative f (a): x3 a3 f (a) = lim . xa x a The cubic in the numerator vanishes for x = a, so it has a factor x a, and in fact the factorization is x3 a3 = (x a)(x2 + x a + a2 ). [Check by expanding!] This gives f (a) = lim (x a)(x2 + x a + a2 ) = lim (x2 + x a + a2 ) = 3a2 . xa xa xa
That is, (x3 ) = 3x2 , in agreement with the Power Rule above for n = 3. Proof of the Power Rule (method 1) The key step is the factorization
xn an = (x a)(xn1 + xn2 a + xn3 a2 + + xan2 + an1 ) This can be checked by expanding the right hand side, distributing the left hand factor: = x(xn1 + xn2 a + xn3 a2 + + xan2 + an1 ) = xn + xn1 a + xn2 a2 + + x2 an2 + xan1 = xn an (x a)(xn1 + xn2 a + xn3 a2 + + xan2 + an1 ) a(xn1 + xn2 a + xn3 a2 + + xan2 + an1 ) xn1 a xn2 a2 xn3 a3 xan1 an
because all the terms in between pair off and cancel out. Much as with x3 , the denition of the derivative gives the derivative of f (x) = xn at x = a as f (a) = xn an xa x a (x a)(xn1 + xn2 a + xn3 a2 + + an1 ) = lim xa xa n1 n2 n3 = lim (x +x a+x a2 + + an1 ) lim =a =a
xa n1 n1
+ an2 a + an3 a2 + + an1
= nan1 ,
+ an1 + an1 + + an1 (n copies)
so f (x) = (xn ) = nxn1 , as claimed. Constant Multiples, Sums and Differences As with limits, we can build up polynomials from these power functions using constant multiples, and sums differences. And the derivatives of these three basic combinations are as simple as with limits.
CHAPTER 3. DIFFERENTIATION RULES
3.1.3
Theorem (The Constant Multiple Rule). If a differentiable function f is multiplied by a constant c, its derivative is multiplied by the same constant: d d [cf (x)] = c [f (x)], or (cf ) (x) = cf (x). dx dx Theorem (The Sum Rule). The sum of two differentiable functions f and g is differentiable, with the sums derivative the sum of summands derivatives: d d d [f (x) + g(x)] = f (x) + g(x), or (f + g) (x) = f (x) + g (x). dx dx dx Theorem (The Difference Rule). The difference of two differentiable functions f and g is differentiable, with its derivative the difference of their derivatives: d d d [f (x) g(x)] = f (x) g(x), or (f g) (x) = f (x) g (x). dx dx dx Warning: These are the only three rules that are as simple and guessable as for limits! Example (5, page 177). Compute the derivative of x8 + 12x5 4x2 + 10x3 6x + 5. The same approach works for differentiating any polynomial. By the way, this shows that all polynomials are differentiable. Example (6, page 178). Find the points on the curve y = x4 6x2 + 4 where the tangent is horizontal. Derivatives of Other Power Functions 1 d 1/2 Example 3 in Section 2.8 show that x has derivative . That is, x = (1/2)x1/21 . This dx 2x ts the power rule, but for power 1/2, not a positive integer. In fact, the rule works for all real powers: Theorem (The Power Rule, Generalized Version). For any real number a, da x = axa1 , or (xa ) = axa1 . dx This is most easily shown later when we know how to differentiate exponential functions and compositions of functions. Example (A). Differentiate 3/x, and use this to check the slope computed in Example 2 of Section 2.7. Derivatives of the Natural Exponential Function In Section 1.5 (page 13 of these notes, page 6 of the text) we dened the number e so that the slope of y = ex at point (0, 1) is 1. That is eh 1 lim = 1. h0 h
CHAPTER 3. DIFFERENTIATION RULES This choic...