Derivative of log42x/x+1

1. Differentiate $1 + \frac{\log{\left(42 x \right)}}{x}$ term by term:

   1. Apply the quotient rule, which is:

      $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

      $f{\left(x \right)} = \log{\left(42 x \right)}$ and $g{\left(x \right)} = x$.

      To find $\frac{d}{d x} f{\left(x \right)}$:

      1. Let $u = 42 x$.

      2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 42 x$:

         1. The derivative of a constant times a function is the constant times the derivative of the function.

            1. Apply the power rule: $x$ goes to $1$

            So, the result is: $42$

         The result of the chain rule is:

         $$\frac{1}{x}$$

      To find $\frac{d}{d x} g{\left(x \right)}$:

      1. Apply the power rule: $x$ goes to $1$

      Now plug in to the quotient rule:

      $\frac{1 - \log{\left(42 x \right)}}{x^{2}}$

   2. The derivative of the constant $1$ is zero.

   The result is: $\frac{1 - \log{\left(42 x \right)}}{x^{2}}$

The answer is:

$$\frac{1 - \log{\left(42 x \right)}}{x^{2}}$$