Derivatives of a fraction functionAn example of a fraction function is: y = − 8 x ( x 2 + 3 ) 2 The quotient rule says that if the function one wishes to differentiate, f ( x ), can be written as: h ( x ) = f ( x ) g ( x ) Then the derivative is (according to what I learned): h ′ ( x ) = f ′ ( x ) g ( x ) − f ( x ) g ′ ( x ) ( g ( x ) ) 2 Then I think the procedure is the following: y ′ = 24 ( x 2 − 1 ) ( ( x 2 + 3 ) 2 ) 2 = 24 ( x 2 − 1 ) ( x 2 + 3 ) 4 However, the solution is... y ′ = 24 ( x 2 − 1 ) ( x 2 + 3 ) 3 What are my mistakes?What is the correct way to derivate fractions?

Question

Derivatives of a fraction functionAn example of a fraction function is:  y  =            −      8      x              (              x        2            +      3              )        2            The quotient rule says that if the function one wishes to differentiate,   f  (  x  ), can be written as:  h  (  x  )  =            f      (      x      )              g      (      x      )      Then the derivative is (according to what I learned):      h    ′    (  x  )  =                    f        ′            (      x      )      g      (      x      )      −      f      (      x      )              g        ′            (      x      )              (      g      (      x      )              )        2            Then I think the procedure is the following:                              y          ′                                    =                              24            (                          x              2                        −            1            )                                (            (                          x              2                        +            3                          )              2                                      )              2                                                                        =                              24            (                          x              2                        −            1            )                                (                          x              2                        +            3                          )              4                                          However, the solution is...      y    ′    =            24      (              x        2            −      1      )              (              x        2            +      3              )        3            What are my mistakes?What is the correct way to derivate fractions?

Jayvion Tyler · Accepted Answer

Your second step (after writing down the quotient rule) should be:      y    ′    =            −      8      (              x        2            +      3              )        2            +      8      x      ⋅      2      ⋅      2      x      (              x        2            +      3      )              (      (              x        2            +      3              )        2                    )        2              ,and then an       x    2    +  3 cancels off and gives you the correct answer:            −      8      (              x        2            +      3              )        2            +      8      x      ⋅      2      ⋅      2      x      (              x        2            +      3      )              (      (              x        2            +      3              )        2                    )        2              =            −      8      (              x        2            +      3      )      +      32              x        2                    (              x        2            +      3              )        3              =            24      (              x        2            −      1      )              (              x        2            +      3              )        3              .

Rebecca Villa · Accepted Answer

Hint #1: If you write your fraction as  H  I  G  H      /    L  O  W  ,then the derivative for the quotient is &quot;given&quot; by the mnemonic            L      O      W      ⋅              d        (        H        I        G        H        )            −      H      I      G      H      ⋅              d        (        L        O        W        )                    (      L      O      W              )        2              .Hint #2:   d  (  H  I  G  H  )  =  −  8Hint #3:   d  (  L  O  W  )  =  2  (      x    2    +  3  )  (  2  x  )  =  4  x  (      x    2    +  3  )Can you take it from here?