Derivative of (x^2+5)/(x-2)

1. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = x^{2} + 5$ and $g{\left(x \right)} = x - 2$.

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

   1. Differentiate $x^{2} + 5$ term by term:

      1. The derivative of the constant $5$ is zero.

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

      The result is: $2 x$

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

   1. Differentiate $x - 2$ term by term:

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

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

      The result is: $1$

   Now plug in to the quotient rule:

   $\frac{- x^{2} + 2 x \left(x - 2\right) - 5}{\left(x - 2\right)^{2}}$

The answer is:

$$\frac{- x^{2} + 2 x \left(x - 2\right) - 5}{\left(x - 2\right)^{2}}$$