Find the derivative of the function below and simplify the result. g ( x ) = ln ⁡ ( 4 + x 2 x )

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Find the derivative of the function below and simplify the result.   g  (  x  )  =  ln  ⁡  (            4      +              x        2              x    )

h2a2l1i2morz · Accepted Answer

Consider the provided function,  g  (  x  )  =  ln  ⁡  (            4      +              x        2              x    )Use the properties of the logarithm function,  ln  ⁡  (      m    n    )  =  ln  ⁡  m  −  ln  ⁡  nSo, we can write as a form.  g  (  x  )  =  ln  ⁡  (            4      +              x        2              x    )    =  ln  ⁡  (      4    +          x      2        )  −  ln  ⁡  xNow, find the derivative of the above function.      d          d      x        (  g  (  x  )  )  =      d          d      x        (  ln  ⁡  (      4    +          x      2        )  −  ln  ⁡  x  )The formula of the logarithm function      d          d      x        (  ln  ⁡  x  )  =      1    x  Apply the formula in the above function,      g    ′    (  x  )  =      1          4      +              x        2                  d          d      x        (      4    +          x      2        )  −      d          d      x        (  ln  ⁡  x  )    =      1          4      +              x        2              (            1      ⋅      2      x              2              4        +                  x          2                      )  −      1    x      =      1          4      +              x        2              (      x          4      +              x        2              )  −      1    x      =      x          4      +              x        2              −      1    x      =                    x        2            −      4      −              x        2                    x      (      4      +              x        2            )          =            −      4              x      (      4      +              x        2            )      Hence the derivative of the provided function is   −      4          x      (              x        2            +      4      )

Find the derivative of the function below and simplify the result. g(x)=ln ((sqrt(4+x^2))/(x))

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