Derivative of lnx/x^5+1/5x

1. Differentiate $\frac{x}{5} + \frac{\log{\left(x \right)}}{x^{5}}$ term by term:

   1. Apply the quotient rule, which is:

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

      $f{\left(x \right)} = \log{\left(x \right)}$ 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. 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: $\frac{1}{5}$

   The result is: $\frac{1}{5} + \frac{- 5 x^{4} \log{\left(x \right)} + x^{4}}{x^{10}}$

2. Now simplify:

   $$\frac{x^{6} - 25 \log{\left(x \right)} + 5}{5 x^{6}}$$

The answer is:

$$\frac{x^{6} - 25 \log{\left(x \right)} + 5}{5 x^{6}}$$