Chain, Product & Quotient Rules
    • The product rule
    • The quotient rule
    • The chain rule
    • Questions
  VCE Maths Methods - Chain, Product & Quotient Rules              2
    The chain rule
    • The chain rule is used to differentiate a function that has a function within 
      it.
      y =f(u) u=f(x)                              3
                                       y =(2x+4)
                                      y =u3 andu=2x+4
                                       
      dy   dy du                      dy =3u2      du =2
      dx = du ⋅dx                       du           dx
      
                                      dy       2                    2
                                      dx =3u ×2=2×3(2x+4)
                                        
                                      dy              2
                                      dx =6(2x+4)
                                        
  VCE Maths Methods - Chain, Product & Quotient Rules                    3
    The product rule
    • The product rule is used to differentiate a function that is the 
     multiplication of two functions.
     f(x)=u(x)×v(x)        dy =⎛du×v⎞+⎛dv ×u⎞ f'(x)=u'v+v'u
                                ⎜      ⎟ ⎜       ⎟   
                           dx   ⎝dx    ⎠ ⎝dx     ⎠
    f(x)=(3x−5)×(4x+7)
      
    u=3x−5          v =4x+7
                     
       u'=3           v'=4
                       
   f '(x)=3(4x+7)+4(3x−5)
     
   f '(x)=12x+21+12x−20=24x+1
     
   f '(x)=24x+1
     
  VCE Maths Methods - Chain, Product & Quotient Rules                4
                The quotient rule
                • The quotient rule is used to differentiate a function that is the division of 
                      two functions.
                      f(x)=u(x)                                                                                     f '(x)=u'v −v'u                                                                   dy =⎛du×v⎞+⎛dv ×u⎞
                                                                                                                                                                        2                                                    ⎜                                ⎟ ⎜                                          ⎟
                                                    v(x)                                                                                                          v                                  dx                      ⎝dx                              ⎠ ⎝dx                                        ⎠
                  f(x)=3x−5
                                                 4x+7
                 u=3x−5                                                                v =4x+7
                        u'=3                                                                v'=4
                                                                                             
               f '(x)= 3(4x+7)−4(3x−5)=12x+21−12x+20
                                                                                                                   2                                                                                                             2
                                                                            (4x+7)                                                                                                        (4x+7)
                    f '(x)=                                         41
                                                                                           2
                                                     (4x+7)
       VCE Maths Methods - Chain, Product & Quotient Rules                                                                                                                                                                                                                                                           5