4100 AWL/Thomas_ch03p147-243 8/19/04 11:16 AM Page 159 3.2 Differentiation Rules 159 3.2 Differentiation Rules This section introduces a few rules that allow us to differentiate a great variety of func- tions. By proving these rules here, we can differentiate functions without having to apply the definition of the derivative each time. Powers, Multiples, Sums, and Differences The first rule of differentiation is that the derivative of every constant function is zero. RULE 1 Derivative of a Constant Function If ƒ has the constant value ƒsxd = c, then dƒ d = scd = 0. dx dx EXAMPLE 1 y If ƒ has the constant value ƒsxd = 8, then c (x, c) (x ϩ h, c) df d ϭ = s8d = 0. y c dx dx Similarly, h d p d x - = 0 and 23 = 0. 0 x x ϩ h dx a 2 b dx a b FIGURE 3.8 The rule sd dxdscd = 0 is Proof of Rule 1 We apply the definition of derivative to ƒsxd = c, the function whose > another way to say that the values of outputs have the constant value c (Figure 3.8). At every value of x, we find that constant functions never change and that the slope of a horizontal line is zero at ƒsx + hd - ƒsxd c - c ƒ¿sxd = lim = lim = lim 0 = 0. every point. h:0 h h:0 h h:0 4100 AWL/Thomas_ch03p147-243 8/19/04 11:16 AM Page 160 160 Chapter 3: Differentiation The second rule tells how to differentiate xn if n is a positive integer. RULE 2 Power Rule for Positive Integers If n is a positive integer, then d - xn = nxn 1 . dx To apply the Power Rule, we subtract 1 from the original exponent (n) and multiply the result by n. EXAMPLE 2 Interpreting Rule 2 ƒ x x2 x3 x4 Á ƒ¿ 12x 3x2 4x3 Á HISTORICAL BIOGRAPHY First Proof of Rule 2 The formula Richard Courant n n n-1 n-2 Á n-2 n-1 (1888–1972) z - x = sz - xdsz + z x + + zx + x d can be verified by multiplying out the right-hand side. Then from the alternative form for the definition of the derivative, ƒszd - ƒsxd zn - xn ƒ ¿sxd = lim - = lim - z:x z x z:x z x n-1 n-2 Á n-2 n-1 = limsz + z x + + zx + x d z:x = nxn-1 Second Proof of Rule 2 If ƒsxd = xn , then ƒsx + hd = sx + hdn . Since n is a positive integer, we can expand sx + hdn by the Binomial Theorem to get ƒsx + hd - ƒsxd sx + hdn - xn ƒ ¿sxd = lim = lim h:0 h h:0 h - nsn - 1d - - xn + nxn 1h + xn 2h2 + Á + nxhn 1 + hn - xn c 2 d = lim h:0 h - nsn - 1d - - nxn 1h + xn 2h2 + Á + nxhn 1 + hn 2 = lim h:0 h - n-1 nsn 1d n-2 Á n-2 n-1 = lim nx + x h + + nxh + h h:0 c 2 d = nxn-1 The third rule says that when a differentiable function is multiplied by a constant, its derivative is multiplied by the same constant. 4100 AWL/Thomas_ch03p147-243 8/19/04 11:16 AM Page 161 3.2 Differentiation Rules 161 RULE 3 Constant Multiple Rule If u is a differentiable function of x, and c is a constant, then d du scud = c . dx dx y In particular, if n is a positive integer, then y ϭ 3x2 d - scxnd = cnxn 1 . dx Slope ϭ 3(2x) EXAMPLE 3 ϭ 6x (a) The derivative formula 3 (1, 3) Slope ϭ 6(1) ϭ 6 d 2 # 2 s3x d = 3 2x = 6x y ϭ x dx 2 says that if we rescale the graph of y = x2 by multiplying each y-coordinate by 3, then we multiply the slope at each point by 3 (Figure 3.9). Slope ϭ 2x (b) A useful special case 1 Slope ϭ 2(1) ϭ 2 (1, 1) The derivative of the negative of a differentiable function u is the negative of the func- tion’s derivative. Rule 3 with c =-1 gives d - = d - # =- # d =-du x s ud s 1 ud 1 sud . 0 1 2 dx dx dx dx FIGURE 3.9 The graphs of y = x2 and Proof of Rule 3 y = 3x2 . Tripling the y-coordinates triples d cusx + hd - cusxd Derivative definition the slope (Example 3). cu = lim dx h:0 h with ƒsxd = cusxd usx + hd - usxd = c lim Limit property h:0 h du = c u is differentiable. dx The next rule says that the derivative of the sum of two differentiable functions is the sum of their derivatives. Denoting Functions by u and Y The functions we are working with when we need a differentiation formula are likely to be denoted by letters like ƒ RULE 4 Derivative Sum Rule and g. When we apply the formula, we + do not want to find it using these same If u and y are differentiable functions of x, then their sum u y is differentiable letters in some other way. To guard at every point where u and y are both differentiable. At such points, against this problem, we denote the d du dy functions in differentiation rules by su + yd = + . letters like u and y that are not likely to dx dx dx be already in use. 4100 AWL/Thomas_ch03p147-243 8/19/04 11:16 AM Page 162 162 Chapter 3: Differentiation EXAMPLE 4 Derivative of a Sum y = x4 + 12x dy d d = sx4d + s12xd dx dx dx = 4x3 + 12 Proof of Rule 4 We apply the definition of derivative to ƒsxd = usxd + ysxd: d [usx + hd + ysx + hd] - [usxd + ysxd] [usxd + ysxd] = lim dx h:0 h usx + hd - usxd ysx + hd - ysxd = lim + h:0 c h h d usx + hd - usxd ysx + hd - ysxd du dy = lim + lim = + . h:0 h h:0 h dx dx Combining the Sum Rule with the Constant Multiple Rule gives the Difference Rule, which says that the derivative of a difference of differentiable functions is the difference of their derivatives. d d du dy du dy su - yd = [u + s -1dy] = + s -1d = - dx dx dx dx dx dx The Sum Rule also extends to sums of more than two functions, as long as there are only finitely many functions in the sum. If u1 , u2 , Á , un are differentiable at x, then so is Á u1 + u2 + + un , and d du1 du2 dun su + u + Á + u d = + + Á + . dx 1 2 n dx dx dx EXAMPLE 5 Derivative of a Polynomial 4 y = x3 + x2 - 5x + 1 3 dy d d 4 d d = x3 + x2 - s5xd + s1d dx dx dx a3 b dx dx 4 = 3x2 + # 2x - 5 + 0 3 8 = 3x2 + x - 5 3 Notice that we can differentiate any polynomial term by term, the way we differenti- ated the polynomial in Example 5. All polynomials are differentiable everywhere. Proof of the Sum Rule for Sums of More Than Two Functions We prove the statement d du1 du2 dun su + u + Á + u d = + + Á + dx 1 2 n dx dx dx by mathematical induction (see Appendix 1). The statement is true for n = 2, as was just proved. This is Step 1 of the induction proof. 4100 AWL/Thomas_ch03p147-243 8/20/04 9:12 AM Page 163 3.2 Differentiation Rules 163 Step 2 is to show that if the statement is true for any positive integer n = k, where k Ú n0 = 2, then it is also true for n = k + 1. So suppose that d du1 du2 duk su + u + Á + u d = + + Á + . (1) dx 1 2 k dx dx dx Then d Á (u + u + + u + u + ) dx (++++)++++*1 2 k ()*k 1 Call the function Call this defined by this sum u. function y. d Á duk+1 d = su + u + + u d + Rule 4 for su + yd dx 1 2 k dx dx du1 du2 Á duk duk+1 = + + + + . Eq. (1) dx dx dx dx With these steps verified, the mathematical induction principle now guarantees the Sum Rule for every integer n Ú 2. EXAMPLE 6 Finding Horizontal Tangents 4 2 y y ϭ x4 Ϫ 2x2 ϩ 2 Does the curve y = x - 2x + 2 have any horizontal tangents? If so, where? Solution The horizontal tangents, if any, occur where the slope dy dx is zero. We have, > dy d (0, 2) = sx4 - 2x2 + 2d = 4x3 - 4x. dx dx dy 1 Now solve the equation = 0 for x: (–1, 1) (1, 1) dx 4 x3 - 4x = 0 x –1 0 1 4 xsx2 - 1d = 0 x = 0, 1, -1. FIGURE 3.10 The curve y = x4 - 2x2 + 2 and its horizontal The curve y = x4 - 2x2 + 2 has horizontal tangents at x = 0, 1, and -1. The corre- tangents (Example 6). sponding points on the curve are (0, 2), (1, 1) and s -1, 1d. See Figure 3.10. Products and Quotients While the derivative of the sum of two functions is the sum of their derivatives, the deriva- tive of the product of two functions is not the product of their derivatives. For instance, d d d d sx # xd = sx2d = 2x, while sxd # sxd = 1 # 1 = 1. dx dx dx dx The derivative of a product of two functions is the sum of two products, as we now explain. RULE 5 Derivative Product Rule If u and y are differentiable at x, then so is their product uy, and d dy du suyd = u + y .