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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 ...

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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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...Product rule quotient wehave seen that the derivative of a sum is derivatives d dx onemightexpectfromthisthatthederivativeofaproductistheproductofthederivatives this not case however in fact it usually happens table contents for instance x dxdx instead nding as follows page functions f and g back print version words rst times second plus home example sinx xsinx xcosx with p says so we verify by establishing equation using denition h lim limg find following ex b t tcost solution e here need to use before dt cost ln sint extends naturally handle any number factors obtained taking one factor at time leaving other unchanged then summing results veried repeatedly see exercise excosx cosx dxcosx exsinx next get...

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