20. Product rule, Quotient rule
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1 20. Prouct rule, Prouct rule Prouct rule, Prouct rule We have seen that the erivative of a sum is the sum of the erivatives: [f(x) + g(x)] = x x [f(x)] + x [(g(x)]. One might expect from this that the erivative of a prouct is the prouct of the erivatives. This is not the case, however. In fact, it usually happens that x [f(x)g(x)] x [f(x)] x [g(x)]. For instance, x [xx] = [ x 2 ] = 2x 1 = (1)(1) = x x [x] x [x]. Instea, the rule for fining the erivative of a prouct is as follows: Prouct rule. For functions f an g, x [f(x)g(x)] = [f(x)] g(x) + f(x) x x [g(x)]. Page 1 of 10 In wors, the erivative of a prouct is the erivative of the first times the secon plus the first times the erivative of the secon.
![In fact, it usually happens that x [f(x)g(x)] x [f(x)] x [g(x)]. For instance, x [xx] = [ x 2 ] = 2x 1 = (1)(1) = x x [x] x [x].](/docs-images/46/21571706/images/page_1.jpg "Instea, the rule for fining the erivative of a prouct is as follows: Prouct rule. For functions f an g, x [f(x)g(x)] = [f(x)] g(x) + f(x) x x [g(x)].")
2 For example, [ x 3 sin x ] = [ x 3 ] sin x + x 3 x x x = 3x 2 sin x + x 3 cos x. [sin x] Prouct rule, Prouct rule With p(x) = f(x)g(x), the rule says that p (x) = f (x)g(x) + f(x)g (x), so we verify the rule by establishing this equation using the efinition of the erivative: p (x) = lim h 0 p(x + h) p(x) h = lim h 0 f(x + h)g(x + h) f(x)g(x) h f(x + h)g(x + h) f(x)g(x + h) + f(x)g(x + h) f(x)g(x) = lim h 0 h ( ) f(x + h) f(x) g(x + h) g(x) = lim g(x + h) + f(x) h 0 h h ( ) ( ) f(x + h) f(x) g(x + h) g(x) = lim lim g(x + h) + f(x) lim h 0 h h 0 h 0 h = f (x)g(x) + f(x)g (x) Example Fin the erivatives of the following functions: Page 2 of 10 (a) f(x) = (x 8 + 2x 3)e x. (b) f(t) = 5 t cos t + 4t 2. Solution
3 (a) f (x) = [ (x 8 + 2x 3)e x] x = [ x 8 + 2x 3 ] e x + (x 8 + 2x 3) x x [ex ] = (8x 7 + 2)e x + (x 8 + 2x 3)e x = (x 8 + 8x 7 + 2x 1)e x. Prouct rule, Prouct rule (b) Here, we nee to use the sum rule before using the prouct rule: f (t) = [ 5 t cos t + 4t 2] t = [ 5 t cos t ] + [ 4t 2 ] t t = [ 5 t ] cos t + 5 t [cos t] + 8t t t = (5 t ln 5) cos t + 5 t ( sin t) + 8t = 5 t (ln 5) cos t 5 t sin t + 8t. Page 3 of 10 The prouct rule extens naturally to hanle any number of factors. For instance, x [f(x)g(x)h(x)] = [f(x)]g(x)h(x) + f(x) [g(x)] h(x) + f(x)g(x) x x x [h(x)].
![4t 2 ] t t = [ 5 t ] cos t + 5 t [cos t] + 8t t t = (5 t ln 5) cos t + 5 t ( sin t) + 8t = 5 t (ln 5) cos t 5 t sin t + 8t.](/docs-images/46/21571706/images/page_3.jpg "Page 3 of 10 The prouct rule extens naturally to hanle any number of factors.")
4 The erivative is obtaine by taking the erivative of one factor at a time, leaving the other factors unchange, an then summing the results. This rule is verifie by using the prouct rule repeately (see Exercise 20 3) Example Fin the erivative of f(x) = (x 3 4x 2 )e x cos x. Prouct rule, Prouct rule Solution f (x) = [ (x 3 4x 2 )e x cos x ] x = x [ x 3 4x 2] e x cos x + (x 3 4x 2 ) x [ex ] cos x + (x 3 4x 2 )e x [cos x] x = (3x 2 8x)e x cos x + (x 3 4x 2 )e x cos x + (x 3 4x 2 )e x ( sin x) = (x 3 x 2 8x)e x cos x (x 3 4x 2 )e x sin x Next, we get the rule for fining the erivative of a quotient. Page 4 of 10
![Prouct rule, Prouct rule Solution f (x) = [ (x 3 4x 2 )e x cos x ] x = x [ x 3 4x 2] e x cos x + (x 3 4x 2 ) x [ex ] cos x + (x 3 4x 2 )e x [cos x] x = (3x](/docs-images/46/21571706/images/page_4.jpg "2 8x)e x cos x + (x 3 4x 2 )e x cos x + (x 3 4x 2 )e x ( sin x) = (x 3 x 2 8x)e x cos x (x 3 4x 2 )e x sin x. 20.2. Next, we get the rule for fining the erivative of a quotient.")
5 . For functions f an g, x [ ] f(x) g(x) [f(x)] f(x) = x x [g(x)] g(x) (g(x)) 2. Prouct rule, Prouct rule In wors, the erivative of a quotient is the bottom times the erivative of the top minus the top times the erivative of the bottom, over the bottom square. The verification (omitte) is very similar to that for the prouct rule Example Fin the erivatives of the following functions: (a) f(x) = x4 2x x 7. x (b) f(t) = 3 sin t t 2 e t. Solution Page 5 of 10
6 (a) f (x) = [ x 4 2x 3 ] + 8 x x 7 x (x 7 x) [ x 4 2x ] (x 4 2x 3 + 8) [ x 7 x ] = x x (x 7 x) 2 = (x7 x)(4x 3 6x 2 ) (x 4 2x 3 + 8)(7x 6 1) (x 7 x) 2. Prouct rule, Prouct rule (b) f (t) = t [ ] 3 sin t t 2 e t (t 2 e t ) = t [3 sin t] (3 sin t) t (t 2 e t ) 2 [ t 2 e t] = (t2 e t )(3 cos t) (3 sin t)(2t e t ) (t 2 e t ) 2. Page 6 of 10 Sometimes a quotient to be ifferentiate can be rewritten in such a way that the quotient rule becomes unnecessary. In this case, going ahea an rewriting is usually preferable to using the quotient rule; the quotient rule shoul be use only if it cannot be avoie.
7 Example Fin the erivative of the function f(x) = x3 5x + 4 x. x Solution At the appropriate step, the function is rewritten in orer to avoi using the quotient rule: f (x) = [ x 3 5x + 4 ] x x x = [x x 1/2] x = 2x 2x 3/2 = 2x 2 ( x) 3. Prouct rule, Prouct rule Page 7 of 10
![to avoi using the quotient rule: f (x) = [ x 3 5x + 4 ] x x x = [x 2 5](/docs-images/46/21571706/images/page_7.jpg "+ 4x 1/2] x = 2x 2x 3/2 = 2x 2 ( x) 3.")
8 20 Exercises Prouct rule, Prouct rule 20 1 Let f(x) = (x 2 3x + 1)(x 4 + 9x 2 ). (a) Fin the erivative of f by first expaning the right-han sie so as to avoi using the prouct rule. (b) Fin the erivative of f by using the prouct rule an verify that the result is the same as that obtaine in part (a) Fin the erivatives of the following functions: (a) f(x) = (2x 5 3x 2 + 1) sin x (b) f(t) = 3 te t 5 t Page 8 of 10
9 20 3 Verify the formula x [f(x)g(x)h(x)] = [f(x)]g(x)h(x) + f(x) [g(x)] h(x) + f(x)g(x) x x x [h(x)]. Prouct rule, Prouct rule Hint: Apply the prouct rule viewing f(x)g(x)h(x) as a prouct with the two factors f(x) an g(x)h(x) Fin the erivative of f(x) = (2x x)8 x cos x. Hint: Use the formula in Exercise Let f(x) = x4 + 3 x 5. x (a) Fin the erivative of f by first rewriting the right-han sie so as to avoi using the quotient rule. (b) Fin the erivative of f by using the quotient rule an verify that the result is the same as that obtaine in part (a). Page 9 of 10
10 20 6 Fin the erivatives of the following functions: (a) f(x) = 3x2 5x 7 (b) f(t) = 4 cos t e t Prouct rule, Prouct rule Page 10 of 10
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