Differentiating logarithmsI am trying to prove that f ( x ) = a l o g ( x ) =&gt; f ′ ( x ) = 1 l n ( a ) ∗ x So I start at f ( x ) = a l o g ( x )Then I move to: f ( x ) = l n ( x ) l n ( a ) And there I get stuck: I want to use the quotient rule, but the entire internet tells me to use the chain rule. And indeed, with the quotient rule I get stuck on an island far away. But still: to me this looks a lot more like: f ( x ) = g ( x ) h ( x ) Then f ( x ) = g ( h ( x ) )So why do I need to use the chain rule from here? How can I use it in this situation?

Question

Differentiating logarithmsI am trying to prove that  f  (  x  )      =    a    l  o  g  (  x  )  =&amp;gt;      f    ′    (  x  )  =      1          l      n      (      a      )      ∗      x      So I start at  f  (  x  )      =    a    l  o  g  (  x  )Then I move to:  f  (  x  )  =            l      n      (      x      )              l      n      (      a      )      And there I get stuck: I want to use the quotient rule, but the entire internet tells me to use the chain rule. And indeed, with the quotient rule I get stuck on an island far away. But still: to me this looks a lot more like:  f  (  x  )  =            g      (      x      )              h      (      x      )      Then  f  (  x  )  =  g  (  h  (  x  )  )So why do I need to use the chain rule from here? How can I use it in this situation?

jugf5 · Accepted Answer

No need for the quotient rule:             1              ln        ⁡        (        a        )             is a constant.  f  (  x  )  =            ln      ⁡      (      x      )              ln      ⁡      (      a      )        =      1          ln      ⁡      (      a      )        ⋅  ln  ⁡  (  x  )    ⟹        f    ′    (  x  )  =            1              ln        ⁡        (        a        )              ⋅      1    x    =      1          ln      ⁡      (      a      )      x