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20. Product rule, Quotient rule Product rule, Quotient rule Product rule 20.1. Product rule Quotient rule Wehave seen that the derivative of a sum is the sum of the derivatives: d [f(x)+g(x)] = d [f(x)]+ d [(g(x)]. dx dx dx Onemightexpectfromthisthatthederivativeofaproductistheproductofthederivatives. This is not the case, however. In fact, it usually happens that d [f(x)g(x)] 6= d [f(x)] d [g(x)]. Table of Contents dx dx dx For instance, ◭◭ ◮◮ d d 2 d d dx [xx] = dx x =2x6=1=(1)(1)= dx[x]dx[x]. ◭ ◮ Instead, the rule for finding the derivative of a product is as follows: Page 1 of 10 Product rule. For functions f and g, Back d [f(x)g(x)] = d [f(x)]g(x)+f(x) d [g(x)]. dx dx dx Print Version In words, the derivative of a product is the derivative of the first times the second plus the Home Page first times the derivative of the second. For example, Product rule, Quotient rule d 3 d 3 3 d Product rule dx x sinx = dx x sinx+x dx[sinx] Quotient rule =3x2sinx+x3cosx. ′ ′ ′ 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 definition of the derivative: ′ p(x+h)−p(x) p (x) = lim h→0 h = lim f(x+h)g(x+h)−f(x)g(x) Table of Contents h→0 h = lim f(x+h)g(x+h)−f(x)g(x+h)+f(x)g(x+h)−f(x)g(x) h→0 h ◭◭ ◮◮ = lim f(x+h)−f(x)g(x+h)+f(x)g(x+h)−g(x) h→0 h h ◭ ◮ =lim f(x+h)−f(x)limg(x+h)+f(x)lim g(x+h)−g(x) h→0 h h→0 h→0 h Page 2 of 10 =f′(x)g(x)+f(x)g′(x). 20.1.1 Example Find the derivatives of the following functions: Back (a) f(x) = (x8 +2x−3)ex. Print Version (b) f(t) = 5tcost+4t2. Home Page Solution (a) Product rule, Quotient rule ′ d 8 x Product rule f (x) = dx (x +2x−3)e Quotient rule d 8 x 8 d x = dx x +2x−3 e +(x +2x−3)dx[e ] 7 x 8 x =(8x +2)e +(x +2x−3)e 8 7 x =(x +8x +2x−1)e . (b) Here, we need to use the sum rule before using the product rule: ′ d t 2 f (t) = dt 5 cost+4t Table of Contents d t d 2 = dt 5 cost + dt 4t d d ◭◭ ◮◮ = 5t cost+5t [cost] + 8t dt dt ◭ ◮ t t =(5 ln5)cost+5 (−sint)+8t t t =5(ln5)cost−5 sint+8t. Page 3 of 10 Back The product rule extends naturally to handle any number of factors. For instance, Print Version d [f(x)g(x)h(x)] = dx Home Page d [f(x)]g(x)h(x)+f(x) d [g(x)]h(x)+f(x)g(x) d [h(x)]. dx dx dx The derivative is obtained by taking the derivative of one factor at a time, leaving the Product rule, Quotient rule other factors unchanged, and then summing the results. This rule is verified by using the product rule repeatedly (see Exercise 20–3). Product rule Quotient rule 20.1.2 Example Find the derivative of f(x) = (x3 − 4x2)excosx. Solution ′ d 3 2 x f (x) = dx (x −4x )e cosx d 3 2 x 3 2 d x = dx x −4x e cosx+(x −4x )dx[e ]cosx 3 2 x d Table of Contents +(x −4x )e dx[cosx] 2 x 3 2 x 3 2 x =(3x −8x)e cosx+(x −4x )e cosx+(x −4x )e (−sinx) ◭◭ ◮◮ =(x3−x2−8x)excosx−(x3−4x2)exsinx. ◭ ◮ 20.2. Quotient rule Page 4 of 10 Next, we get the rule for finding the derivative of a quotient. Back Print Version Home Page
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