tailieunhanh - Stewart - Calculus - Early Transcendentals 6e HQ (Thomson, 2008) Episode 3

Tham khảo tài liệu 'stewart - calculus - early transcendentals 6e hq (thomson, 2008) episode 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 174 Illi CHAPTERS DIFFERENTIATION RULES Comparing the equations in 1 2 and 3 we see a pattern emerging. It seems to be a reasonable guess that when n is a positive integer d dx x nx This turns out to be true. THE PCWER RULE If n is a positive integer then -7- x nx 1 ax FIRST PROOF The formula x an x tz x 1 x 2ữ xan 2 a 1 can be verified simply by multiplying out the right-hand side or by summing the second factor as a geometric series . If x x we can use Equation for fifa and the equation above to write JW fl . x fl f i lịm 7 c lim ii X a ĨI X a lim x x 2ữ xan 2 tz 1 xl a tz 1 a 2a aan 2 an 1 na 1 SECOND PROOF f x h f x u h n x Mx Inn J lim --------------- ỈÌ 10 z II I 0 fl The Binomial Theorem is given on Reference Page 1. In finding the derivative of X4 we had to expand x z 4. Here we need to expand x z and we use the Binomial Theorem to do so f x x nx lh lim ----------------- h 0 y -x 2 z2 nx z 1 hn h x ỉ n n 1 nx lh -xn 2h2 nx z 1 z lim-------------------------------------------- A I 0 fl Ị lim nx Al 0 nf n x 2h nxhn 2 hn nx 1 because every term except the first has h as a factor and therefore approaches 0. We illustrate the Power Rule using various notations in Example 1. EXAMPLE 1 a If x X6 then y x 6x5. c If V Í4 then 4i3. b Ify X1000 then v lOOOx999. d - r3 3r2 dr SECTION DERIVATIVES OF POLYNOMIALSAND EXPONENTIAL FUNCTIONS Illi 175 What about power functions with negative integer exponents In Exercise 61 we ask you to verify from the definition of a derivative that 1 d M dx X We can rewrite this equation as D 2 dx and so the Power Rule is true when n 1. In fact we will show in the next section Exercise 58 c that it holds for all negative integers. What if the exponent is a fraction In Example 3 in Section we found that 7x1 1 dx. 2-y x which can be written as x1 2 x 1 2 This shows that the Power Rule is true even when n 5. In fact we will show in Section that it is true for all real numbers n. THE PCWER RULE GENERAL VERSON If n is any real

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