Derivative of -(x^2+289)/x

1. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = - x^{2} - 289$ and $g{\left(x \right)} = x$.

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

   1. Differentiate $- x^{2} - 289$ term by term:

      1. The derivative of the constant $-289$ is zero.

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

         1. Apply the power rule: $x^{2}$ goes to $2 x$

         So, the result is: $- 2 x$

      The result is: $- 2 x$

   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:

   $\frac{289 - x^{2}}{x^{2}}$

2. Now simplify:

   $$-1 + \frac{289}{x^{2}}$$

The answer is:

$$-1 + \frac{289}{x^{2}}$$