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:
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                    Product rule. For functions f and g,
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                               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.
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          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.
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           The product rule extends naturally to handle any number of factors. For instance,
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                    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
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