Why can't I use product rule to derive x ln(3)?The product rule is defined as ( f ⋅ g ) ′ = f ′ ⋅ g + g ′ ⋅ f .I have the following function u ( x ) = x ⋅ ln ⁡ ( 3 ). I understand that you can derive it by implicit differentiation and have ln ⁡ ( 3 ) as the result.I, however, do not understand why I get the wrong result by applying the product rule: f ( x ) = x g ( x ) = ln ⁡ ( 3 ) f ′ ( x ) = 1 g ′ ( x ) = 1 / 3 D ( f ( x ) ∗ g ( x ) ) = = 1 ∗ ln ⁡ ( 3 ) + 1 / 3 ∗ x = ln ⁡ ( 3 ) + 1 / 3 x ≠ l n ( 3 )

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Why can&#039;t I use product rule to derive x ln(3)?The product rule is defined as  (  f  ⋅  g      )    ′    =      f    ′    ⋅  g  +      g    ′    ⋅  f  .I have the following function   u  (  x  )  =  x  ⋅  ln  ⁡  (  3  ). I understand that you can derive it by implicit differentiation and have   ln  ⁡  (  3  ) as the result.I, however, do not understand why I get the wrong result by applying the product rule:  f  (  x  )  =  x    g  (  x  )  =  ln  ⁡  (  3  )        f    ′    (  x  )  =  1        g    ′    (  x  )  =  1      /    3    D  (  f  (  x  )  ∗  g  (  x  )  )  =    =  1  ∗  ln  ⁡  (  3  )  +  1      /    3  ∗  x    =  ln  ⁡  (  3  )  +  1      /    3  x  ≠  l  n  (  3  )

podesect · Accepted Answer

ln  ⁡  (  3  ) is a constant.. There&#039;s no need for the product rule.  (  x  ⋅  c      )    ′    =  c when   c is a constant.You can use the product rule, but there&#039;s no need for it, since the derivative of any constant   c is given by   (  c      )    ′    =  0. That gives us   (  ln  ⁡  3      )    ′    =  0If   x  =  f  (  x  ) and   ln  ⁡  3  =  g  (  x  ), then the derivative, using the product rule, is given by:      f    ′    (  x  )  g  (  x  )  +  f  (  x  )      g    ′    (  x  )  =  (  1  )  ⋅  ln  ⁡  (  3  )  +  x  ⋅  (  0  )  =  ln  ⁡  (  3  )Note that for   g  (  x  )  =  ln  ⁡  3  ,        g    ′    (  x  )  ≠      1    3  . It is true that if   h  (  x  )  =  ln  ⁡  x  ,   then       h    ′    (  x  )  =      1    x  , but   x is the variable with respect to which we are differentiating. In contrast, the argument of   ln  ⁡  (  3  ),   3, is a constant, as is   ln  ⁡  3, and like any constant or constant function, in this case, its derivative with respect to   x is   0

Brenden Tran · Accepted Answer

It is not necessary to use the productrule (see the answer of amWhy). You can however consider   ln  ⁡  (  3  ) to be a function of   x. It is a constant function. The derivative of any constant function is   0 and applying the productrule with   f  (  x  )  =  x and   g  (  x  )  =  ln  ⁡  (  3  ) gives:      f    ′    (  x  )  g  (  x  )  +  f  (  x  )      g    ′    (  x  )  =  1  ×  ln  ⁡  (  3  )  +  x  ×  0  =  ln  ⁡  3

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