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

Question

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

eminecic2rg · Accepted Answer

2          cot      x        +    2          tan      x       or       2          sec      x              csc      x      , depending on preferred simplificationStep 1No simplification      y    =          ln              (                              tan            2                    x                )            Note that             d              d        x                    ln              (        x        )              =          1      x      , so by the chain rule             d              d        x                    ln              (        u        )              =          1      u        ⋅                  d        u                    d        x            . Next:                    d        y                    d        x              =          1                        tan          2                x              ⋅          d              d        x                            tan        2            x      Because we have a function squared, we must use the chain rule once more:                    d        y                    d        x              =          1                        tan          2                x              ⋅    2          tan      x        ⋅          d              d        x                    tan      x                          d        y                    d        x              =          1                        tan          2                x              ⋅    2          tan      x        ⋅                  sec        2            x                          d        y                    d        x              =                            cos          2                x                              sin          2                x              ⋅                  2                  sin          x                            cos        x              ⋅          1                        cos          2                x                                d        y                    d        x              =          2                        sin          x                          cos          x                                        d        y                    d        x              =    2          sec      x              csc      x      Step 2SimplificationUsing the logarithm rules:            ln              (                  a          b                )              =    b          ln              (        a        )                        ln              (        a        /        b        )              =          ln              (        a        )              -          ln              (        b        )            Then:      y    =          ln              (                              tan            2                    x                )                  y    =          ln              (                                            sin              2                        x                                              cos              2                        x                          )                  y    =          ln              (                              sin            2                    x                )              -          ln              (                              cos            2                    x                )                  y    =    2          ln              (                  sin          x                )              -    2          ln              (                  cos          x                )            Then differentiating becomes easier:                    d        y                    d        x              =    2    ⋅          1              sin        x              ⋅          d              d        x                    sin      x        -    2    ⋅          1              cos        x              ⋅          d              d        x                    cos      x                          d        y                    d        x              =    2          (                        cos          x                          sin          x                    +                        sin          x                          cos          x                    )                          d        y                    d        x              =                  2                  (                                    cos              2                        x                    +                                    sin              2                        x                    )                                      sin          x                          cos          x                                        d        y                    d        x              =          2                        sin          x                          cos          x                                        d        y                    d        x              =    2          sec      x              csc      x

daliner4jl · Accepted Answer

good

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

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