How to find the derivative of ln ( ln x 2 ) ?

Question

How to find the derivative of             ln              (                              ln            x                    2                )             ?

trinijovenvy5 · Accepted Answer

Step 1We have:            d              d        x                    (              ln                  (                                    ln              x                        2                    )                    )      According to the chain rule,             d              d        x                    f              (                  g                      (            x            )                          )              =          d              d        x                    (      f      )        ⋅          d              d        x                    g              (        x        )            Here,             f              (        u        )              =          ln      u       where       u    =                            ln          x                2                  Since             d              d        u                    ln      u        =          1      u       , we now have:            1      u        ⋅          d              d        x                                      ln          x                2                  Here,             f              (        v        )              =          ln      v       where       v    =          x      2       , so we have:            1      u        ⋅          1      v        ⋅    2    x  Since       u    =                            ln          x                2                   and       v    =          x      2       , we have:                    2        x                              ln                      (                          x              2                        )                                    x          2                    Since                               ln          x                n            =        n          ln      x       , we get:                    2        x                    2                  x          2                          ln          x                                1              x                  ln          x

trkshkjf · Accepted Answer

Step 1We can use chain rule here. We can write             f              (        x        )              =          ln              (                              ln            x                    2                )             as            f              (        x        )              =          ln              (                  g                      (            x            )                          )             ,             g              (        x        )              =          ln              (        h                  (          x          )                )             and       h          (      x      )        =          x      2      then                     d        f                    d        g              =          1                        g                      (            x            )                               ,                     d        g                    d        h              =          1              h                  (          x          )                     and                     d        g                    d        h              =    2    x  and using chain rule as                     d        f                              d          x                      =                  d        f                    d        g              ×                  d        g                    d        h              ×                  d        h                              d          x                                1                        g                      (            x            )                                ×          1              h                  (          x          )                      ×    2    x              1                                    ln            x                    2                ⋅                    1              x        2              ⋅    2    x              2              x                              ln            x                    2                    Hence             d              d        x                    ln              (                              ln            x                    2                )              =          2              x                              ln            x                    2                      =          2              x        2                  ln          x                      =          1              x                  ln          x

How do you find the derivative of ln(ln x^2)?

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