Find where f ( x ) = 1 ln ( x ) is continuous and differentiable on which interval(s)?

Question

Find where             f              (        x        )              =          1              ln                  (          x          )                     is continuous and differentiable on which interval(s)?

Devyn Rosario · Accepted Answer

It&#039;s continuous and differentiable on the union             (      0      ,      1      )        ∪          (      1      ,      ∞      )      . That is, for all       x    &amp;gt;    0   except       x    =    1  .Explanation:The function ln(x) is continuous and differentiable for all       x    &amp;gt;    0  . Thus,             1                        ln                      (            x            )                               will be continuous and differentiable for all such values of x as well, except for those values of x where             ln              (        x        )              =    0  . The only such value of x where the logarithm is zero is       x    =    1  .Hence,             1                        ln                      (            x            )                               is continuous and differentiable on the union             (      0      ,      1      )        ∪          (      1      ,      ∞      )      . Thus, for all       x    &amp;gt;    0   except       x    =    1  .Then, if you are interested, the Quotient Rule gives:            d              d        x                    (              1                              ln                          (              x              )                                          )        =                            ln                      (            x            )                          ⋅        0        -        1        ⋅                  1          x                                      (                      ln                          (              x              )                                )                2              =    -          1              x                              (                          ln                              (                x                )                                      )                    2                     for       x    &amp;gt;    0   and       x    ≠    1  .

Find where f(x)=1/(ln(x)) is continuous and differentiable on which interval(s)?

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