Find the derivative of this function using logarithmic differentiation: y = ( x + 3 ) 4 ( 2 x 2 + 5 x ) 3 4 x − 3

Question

Find the derivative of this function using logarithmic differentiation:  y  =                    (        x        +        3                  )          4                (        2                  x          2                +        5        x                  )          3                            4        x        −        3

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From here,      y    ′    =  y  ⋅  (  ln  ⁡  y      )    ′  These log rules will be helpful to you:                    ln        ⁡        (        a        b        )                            =        ln        ⁡        a        +        ln        ⁡        b                            ln        ⁡                  (                                    a              b                                )                                    =        ln        ⁡        a        −        ln        ⁡        b                            ln        ⁡                  a          b                                    =        b        ln        ⁡        a            And                    d                              d                x              ln  ⁡      (    g    (    x    )    )    =                              g          ′                (        x        )                    g        (        x        )            Here is an example of derivating a function using the rules mentioned above.                    y                            =        (        1        +        x        )                              (            1            +                          x              2                        )                    2                                      (            1            +                          x              3                        )                    3                                    ln        ⁡        y                            =        ln        ⁡                  [          (          1          +          x          )                                    (              1              +                              x                2                            )                        2                                              (              1              +                              x                3                            )                        3                    ]                                                  =        ln        ⁡        (        1        +        x        )        +        2        ln        ⁡                  (          1          +                      x            2                    )                +        3        ln        ⁡                  (          1          +                      x            3                    )                                    (        ln        ⁡        y                  )          ′                                    =                              1                          1              +              x                                      +                                            4              x                                      1              +                              x                2                                                    +                                            9                              x                2                                                    1              +                              x                3                                                                                  y          ′                                    =        y        ⋅        (        ln        ⁡        y                  )          ′                =        (        1        +        x        )                              (            1            +                          x              2                        )                    2                                      (            1            +                          x              3                        )                    3                          [                                    1                              1                +                x                                              +                                                    4                x                                            1                +                                  x                  2                                                              +                                                    9                                  x                  2                                                            1                +                                  x                  3                                                              ]

Find the derivative of this function using logarithmic differentiation: y=((x+3)^4(2x^2+5x)^3)/(sqrt(4x−3))

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