Derivative of ln(x)/x^5

1. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = \log{\left(x \right)}$ and $g{\left(x \right)} = x^{5}$.

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

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

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

   1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$

   Now plug in to the quotient rule:

   $\frac{- 5 x^{4} \log{\left(x \right)} + x^{4}}{x^{10}}$

2. Now simplify:

   $$\frac{1 - 5 \log{\left(x \right)}}{x^{6}}$$

The answer is:

$$\frac{1 - 5 \log{\left(x \right)}}{x^{6}}$$