How to find the derivative of y = ln ( 5 x ) ?

Question

How to find the derivative of       y    =          ln              (        5        x        )            ?

Kylie Woodward · Accepted Answer

Suppose,       y    =          ln              (        b                  (          x          )                )            Thus using Chain Rule,      y    ′    =          1              b                  (          x          )                      ⋅          (      b              (        x        )            )        ′  Similarly following for the above function yields,      y    ′    =          1              5        x              ⋅          (      5      x      )        ′        y    ′    =          1              5        x              ⋅    5  Hence,       y    ′    =          1      x

Rodney Hoover · Accepted Answer

A student comfortable with the natural logarithm function and its properties might think of this:
One could reason as follows:

y = \ln (5 x) = \ln (5) + \ln (x)

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But ln(5) is a constant, so its derivative is 0.
Hence,

\frac{d y}{d x} = \frac{d}{d x} (\ln 5 + \ln x) = \frac{d}{d x} (\ln x) = \frac{1}{x}

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How to find the derivative of y = ln ( 5 x ) ?

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